Energy storage controller with battery life model

ABSTRACT

An electrical energy storage system includes a battery configured to store and discharge electric power to an energy grid, a power inverter configured to use battery power setpoints to control an amount of the electric power stored or discharged from the battery, the battery power setpoints comprising at least one of frequency regulation power setpoints and ramp rate control power setpoints, and a controller. The controller is configured to use a battery life model to generate the battery power setpoints for the power inverter. The battery life model includes one or more variables that depend on the battery power setpoints.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/247,793, filed Aug. 25, 2016, now U.S. Pat. No. 10,250,039, whichclaims the benefit of and priority to U.S. Provisional PatentApplication No. 62/239,131, U.S. Provisional Patent Application No.62/239,231, U.S. Provisional Patent Application No. 62/239,233, U.S.Provisional Patent Application No. 62/239,245, U.S. Provisional PatentApplication No. 62/239,246, and U.S. Provisional Patent Application No.62/239,249, each of which has a filing date of Oct. 8, 2015. The entiredisclosure of each of these patent applications is incorporated byreference herein.

BACKGROUND

The present disclosure relates generally to frequency response systemsconfigured to add or remove electricity from an energy grid, and moreparticularly to a frequency response controller that determines optimalpower setpoints for a battery power inverter in a frequency responsesystem.

Increased concerns about environmental issues such as global warminghave prompted an increased interest in alternate clean and renewablesources of energy. Such sources include solar and wind power. One methodfor harvesting solar energy is using a photovoltaic (PV) field whichprovides power to an energy grid supplying regional power.

Availability of solar power depends on the time of day (sunrise andsunsets) and weather variables such as cloud cover. The power output ofa PV field can be intermittent and may vary abruptly throughout thecourse of a day. For example, a down-ramp (i.e., a negative change) inPV output power may occur when a cloud passes over a PV field. Anup-ramp (i.e., a positive change) in PV output power may occur atsunrise and at any time during the day when a cloudy sky above the PVfield clears up. This intermittency in PV power output presents aproblem to the stability of the energy grid. In order to address theintermittency of PV output power, ramp rate control is often used tomaintain the stability of the grid.

Ramp rate control is the process of offsetting PV ramp rates that falloutside of compliance limits determined by the electric power authorityoverseeing the grid. Ramp rate control typically requires the use of anenergy source that allows for offsetting ramp rates by either supplyingadditional power to the grid or consuming more power from the grid.Stationary battery technology can been used for such applications.Stationary battery technology can also be used for frequency regulation,which is the process of maintaining the grid frequency at a desiredvalue (e.g. 60 Hz in the United States) by adding or removing energyfrom the grid as needed. However, it is difficult and challenging toimplement both ramp rate control and frequency regulationsimultaneously.

Ramp rate control and frequency regulation both impact the rate at whichenergy is provided to or removed from the energy grid. However, ramprate control and frequency regulation often have conflicting objectives(i.e., controlling PV ramp rates vs. regulating grid frequency) whichthe same battery from being used for ramp rate control and frequencyregulation simultaneously. Additionally, conventional ramp rate controland frequency regulation techniques can result in premature degradationof battery assets and often fail to maintain the state-of-charge of thebattery within an acceptable range. It would be desirable to providesolutions to these and other disadvantages of conventional ramp ratecontrol and frequency regulation techniques.

SUMMARY

One implementation of the present disclosure is an electrical energystorage system. The system includes a battery configured to store anddischarge electric power to an energy grid, a power inverter configuredto use battery power setpoints to control an amount of the electricpower stored or discharged from the battery, the battery power setpointscomprising at least one of frequency regulation power setpoints and ramprate control power setpoints, and a controller. The controller isconfigured to use a battery life model to generate the battery powersetpoints for the power inverter. The battery life model includes one ormore variables that depend on the battery power setpoints.

In some embodiments, the controller is configured to generate thefrequency regulation power setpoints based on a frequency of the energygrid, generate the ramp rate control power setpoints based on a poweroutput of a photovoltaic field, and combine the frequency regulationpower setpoints and the ramp rate control power setpoints to generatethe battery power setpoints.

In some embodiments, the one or more variables that depend on thebattery power setpoints include at least one of a temperature of thebattery, a state-of-charge of the battery, a depth of discharge of thebattery, a power ratio of the battery, and an effort ratio of thebattery.

In some embodiments, the controller is configured to determine arelationship between each of the variables in of the battery life modeland one or more variables manipulated by the controller.

In some embodiments, the controller is configured to use the batterylife model to estimate an amount of battery degradation that will resultfrom the battery power setpoints.

In some embodiments, the controller is configured to use the estimatedamount of battery degradation to estimate a cost of the batterydegradation that will result from the battery power setpoints.

In some embodiments, the controller is configured to estimate an amountof frequency response revenue that will result from the battery powersetpoints and generate the battery power setpoints as a function of boththe estimated cost of battery degradation and the estimated amount offrequency response revenue that will result from the battery powersetpoints.

In some embodiments, the battery life model is a parametric model. Theparametric model may include a regression coefficient for each of theone or more variables in the battery life model. The controller may beconfigured to perform a curve fitting process to determine values forthe regression coefficients.

Another implementation of the present disclosure is a method foroperating an electrical energy storage system. The method includesstoring electric power in a battery and discharging the stored electricpower to an energy grid. The method may further include using batterypower setpoints to control an amount of the electric power stored ordischarged from the battery. The battery power setpoints may include atleast one of frequency regulation power setpoints and ramp rate controlpower setpoints. The method may further include using a battery lifemodel to generate the battery power setpoints. The battery life modelmay include one or more variables that depend on the battery powersetpoints.

In some embodiments, the method my further include generating thefrequency regulation power setpoints based on a frequency of the energygrid, generating the ramp rate control power setpoints based on a poweroutput of a photovoltaic field, and combining the frequency regulationpower setpoints and the ramp rate control power setpoints to generatethe battery power setpoints.

In some embodiments, the one or more variables that depend on thebattery power setpoints include at least one of a temperature of thebattery, a state-of-charge of the battery, a depth of discharge of thebattery, a power ratio of the battery, and an effort ratio of thebattery.

In some embodiments, the method may further include determining arelationship between each of the variables in of the battery life modeland one or more variables manipulated by a controller.

In some embodiments, the method further includes using the battery lifemodel to estimate an amount of battery degradation that will result fromthe battery power setpoints.

In some embodiments, the method further includes using the estimatedamount of battery degradation to estimate a cost of the batterydegradation that will result from the battery power setpoints.

In some embodiments, the method further includes estimating an amount offrequency response revenue that will result from the battery powersetpoints and generating the battery power setpoints as a function ofboth the estimated cost of battery degradation and the estimated amountof frequency response revenue that will result from the battery powersetpoints.

In some embodiments, the battery life model is a parametric model. Theparametric model includes a regression coefficient for each of the oneor more variables in the battery life model. In some embodiments, themethod further includes performing a curve fitting process to determinevalues for the regression coefficients.

Another implementation of the present disclosure is a controller for anelectrical energy storage system. The controller includes a processingcircuit. The processing circuit is configured to use a battery lifemodel to generate battery power setpoints. The battery life modelcomprising one or more variables that depend on the battery powersetpoints. The processing circuit is further configured to use thebattery power setpoints to control an amount of electric power stored ordischarged from a battery. The battery power setpoints comprising atleast one of frequency regulation power setpoints and ramp rate controlpower setpoints.

In some embodiments, the processing circuit is further configured togenerate the frequency regulation power setpoints based on a frequencyof the energy grid, generate the ramp rate control power setpoints basedon a power output of a photovoltaic field, and combine the frequencyregulation power setpoints and the ramp rate control power setpoints togenerate the battery power setpoints.

In some embodiments, the one or more variables that depend on thebattery power setpoints include at least one of a temperature of thebattery, a state-of-charge of the battery, a depth of discharge of thebattery, a power ratio of the battery, and an effort ratio of thebattery.

In some embodiments, the processing circuit is further configured to usethe battery life model to estimate an amount of battery degradation thatwill result from the battery power setpoints. In some embodiments, theprocessing circuit is further configured to use the estimated amount ofbattery degradation to estimate a cost of the battery degradation thatwill result from the battery power setpoints.

Those skilled in the art will appreciate that the summary isillustrative only and is not intended to be in any way limiting. Otheraspects, inventive features, and advantages of the devices and/orprocesses described herein, as defined solely by the claims, will becomeapparent in the detailed description set forth herein and taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a frequency response optimization system,according to an exemplary embodiment.

FIG. 2 is a graph of a regulation signal which may be provided to thesystem of FIG. 1 and a frequency response signal which may be generatedby the system of FIG. 1, according to an exemplary embodiment.

FIG. 3 is a block diagram illustrating the frequency response controllerof FIG. 1 in greater detail, according to an exemplary embodiment.

FIG. 4 is a block diagram illustrating the high level controller of FIG.3 in greater detail, according to an exemplary embodiment.

FIG. 5 is a block diagram illustrating the low level controller of FIG.3 in greater detail, according to an exemplary embodiment.

FIG. 6 is a flowchart of a process for determining frequency responsemidpoints and battery power setpoints that maintain the battery at thesame state-of-charge at the beginning and end of each frequency responseperiod, according to an exemplary embodiment.

FIG. 7 is a flowchart of a process for determining optimal frequencyresponse midpoints and battery power setpoints in the presence of demandcharges, according to an exemplary embodiment.

FIG. 8A is a block diagram of an electrical energy storage systemconfigured to simultaneously perform both ramp rate control andfrequency regulation while maintaining the state-of-charge of thebattery within a desired range, according to an exemplary embodiment.

FIG. 8B is a drawing of the electrical energy storage system of FIG. 8A,according to an exemplary embodiment.

FIG. 9 is a flowchart of a process for generating and using a batterylife model to control battery power setpoints, according to an exemplaryembodiment.

FIG. 10 is a graph illustrating a reactive ramp rate control techniquewhich can be used by the electrical energy storage system of FIGS.8A-8B, according to an exemplary embodiment.

FIG. 11 is a graph illustrating a preemptive ramp rate control techniquewhich can be used by the electrical energy storage system of FIGS.8A-8B, according to an exemplary embodiment.

FIG. 12 is a block diagram of a frequency regulation and ramp ratecontroller which can be used to monitor and control the electricalenergy storage system of FIGS. 8A-8B, according to an exemplaryembodiment.

FIG. 13 is a block diagram of a frequency response control system,according to an exemplary embodiment.

FIG. 14 is a block diagram illustrating data flow into a data fusionmodule of the frequency response control system of FIG. 13, according toan exemplary embodiment.

FIG. 15 is a block diagram illustrating a database schema which can beused in the frequency response control system of FIG. 13, according toan exemplary embodiment.

DETAILED DESCRIPTION

Referring generally to the FIGURES, systems and methods for controllingand using electrical energy storage are shown, according to variousexemplary embodiments. Electrical energy storage (e.g., a battery) canbe used for several applications, two of which are ramp rate control andfrequency regulation. Ramp rate control is the process of offsettingramp rates (i.e., increases or decreases in the power output of anenergy system such as a photovoltaic energy system) that fall outside ofcompliance limits determined by the electric power authority overseeingthe energy grid. Ramp rate control typically requires the use of anenergy source that allows for offsetting ramp rates by either supplyingadditional power to the grid or consuming more power from the grid. Insome instances, a facility is penalized for failing to comply with ramprate requirements.

Frequency regulation (also referred to as frequency response) is theprocess of maintaining the grid frequency at a desired value (e.g. 60 Hzin the United States) by adding or removing energy from the grid asneeded. During a fluctuation of the grid frequency, a frequencyregulation system may offset the fluctuation by either drawing moreenergy from the energy grid (e.g., if the grid frequency is too high) orby providing energy to the energy grid (e.g., if the grid frequency istoo low). A facility participating in a frequency regulation program mayreceive a regulation signal from a utility or other entity responsiblefor regulating the frequency of the energy grid. In response to theregulation signal, the facility adds or removes energy from the energygrid. The facility may be provided with monetary incentives or awards inexchange for participating in the frequency regulation program.

Storing electrical energy in a battery may allow a facility to performfrequency regulation and/or ramp rate control. However, repeatedlycharging and discharging the battery may cause battery degradation andreduce battery life. In some instances, the costs of battery degradation(e.g., decreased ability to store and discharge energy, reducedpotential for participating in frequency regulation programs, etc.)outweigh the monetary incentives or cost savings that would be gainedfrom using the battery to perform frequency regulation and/or ramp ratecontrol. However, it is difficult and challenging to predict the costsof battery degradation. Additionally, even if the costs of batterydegradation are known, it can be difficult to determine appropriatecontrol actions to preserve battery life.

Advantageously, the systems and methods described herein use a batterylife model to predict the battery degradation that will result fromvarious control actions (e.g., charging or discharging the battery). Thebattery life model may estimate battery capacity loss as a function ofvariables that can be controlled (e.g., by a battery controller) whileperforming ramp rate control and/or frequency regulation. Losses inbattery capacity can be converted into revenue losses and compared withthe potential revenue gains resulting from battery usage (e.g.,frequency response revenue). This allows the controller to make aninformed decision regarding battery usage by considering the tradeoffbetween revenue generation potential and battery degradation.

The following sections of this disclosure describe the battery lifemodel in greater detail as well as several exemplary systems in whichthe battery life model may be used. For example, the battery life modelcan be used in a frequency response optimization system to determineoptimal battery power setpoints while participating in a frequencyresponse program. The frequency response implementation is describedwith reference to FIGS. 1-7. The battery life model can also be used todetermine optimal battery power setpoints and photovoltaic powersetpoints in a photovoltaic (PV) energy system that simultaneouslyperforms both frequency regulation and ramp rate control. The PV energysystem implementation is described with reference to FIGS. 8A-8B.

Frequency Response Optimization

Referring to FIG. 1, a frequency response optimization system 100 isshown, according to an exemplary embodiment. System 100 is shown toinclude a campus 102 and an energy grid 104. Campus 102 may include oneor more buildings 116 that receive power from energy grid 104. Buildings116 may include equipment or devices that consume electricity duringoperation. For example, buildings 116 may include HVAC equipment,lighting equipment, security equipment, communications equipment,vending machines, computers, electronics, elevators, or other types ofbuilding equipment. In some embodiments, buildings 116 are served by abuilding management system (BMS). A BMS is, in general, a system ofdevices configured to control, monitor, and manage equipment in oraround a building or building area. A BMS can include, for example, aHVAC system, a security system, a lighting system, a fire alertingsystem, and/or any other system that is capable of managing buildingfunctions or devices. An exemplary building management system which maybe used to monitor and control buildings 116 is described in U.S. patentapplication Ser. No. 14/717,593, titled “Building Management System forForecasting Time Series Values of Building Variables” and filed May 20,2015, the entire disclosure of which is incorporated by referenceherein.

In some embodiments, campus 102 includes a central plant 118. Centralplant 118 may include one or more subplants that consume resources fromutilities (e.g., water, natural gas, electricity, etc.) to satisfy theloads of buildings 116. For example, central plant 118 may include aheater subplant, a heat recovery chiller subplant, a chiller subplant, acooling tower subplant, a hot thermal energy storage (TES) subplant, anda cold thermal energy storage (TES) subplant, a steam subplant, and/orany other type of subplant configured to serve buildings 116. Thesubplants may be configured to convert input resources (e.g.,electricity, water, natural gas, etc.) into output resources (e.g., coldwater, hot water, chilled air, heated air, etc.) that are provided tobuildings 116. An exemplary central plant which may be used to satisfythe loads of buildings 116 is described U.S. patent application Ser. No.14/634,609, titled “High Level Central Plant Optimization” and filedFeb. 27, 2015, the entire disclosure of which is incorporated byreference herein.

In some embodiments, campus 102 includes energy generation 120. Energygeneration 120 may be configured to generate energy that can be used bybuildings 116, used by central plant 118, and/or provided to energy grid104. In some embodiments, energy generation 120 generates electricity.For example, energy generation 120 may include an electric power plant,a photovoltaic energy field, or other types of systems or devices thatgenerate electricity. The electricity generated by energy generation 120can be used internally by campus 102 (e.g., by buildings 116 and/orcampus 118) to decrease the amount of electric power that campus 102receives from outside sources such as energy grid 104 or battery 108. Ifthe amount of electricity generated by energy generation 120 exceeds theelectric power demand of campus 102, the excess electric power can beprovided to energy grid 104 or stored in battery 108. The power outputof campus 102 is shown in FIG. 1 as P_(campus). P_(campus) may bepositive if campus 102 is outputting electric power or negative ifcampus 102 is receiving electric power.

Still referring to FIG. 1, system 100 is shown to include a powerinverter 106 and a battery 108. Power inverter 106 may be configured toconvert electric power between direct current (DC) and alternatingcurrent (AC). For example, battery 108 may be configured to store andoutput DC power, whereas energy grid 104 and campus 102 may beconfigured to consume and generate AC power. Power inverter 106 may beused to convert DC power from battery 108 into a sinusoidal AC outputsynchronized to the grid frequency of energy grid 104. Power inverter106 may also be used to convert AC power from campus 102 or energy grid104 into DC power that can be stored in battery 108. The power output ofbattery 108 is shown as P_(bat). P_(bat) may be positive if battery 108is providing power to power inverter 106 or negative if battery 108 isreceiving power from power inverter 106.

In some instances, power inverter 106 receives a DC power output frombattery 108 and converts the DC power output to an AC power output thatcan be fed into energy grid 104. Power inverter 106 may synchronize thefrequency of the AC power output with that of energy grid 104 (e.g., 50Hz or 60 Hz) using a local oscillator and may limit the voltage of theAC power output to no higher than the grid voltage. In some embodiments,power inverter 106 is a resonant inverter that includes or uses LCcircuits to remove the harmonics from a simple square wave in order toachieve a sine wave matching the frequency of energy grid 104. Invarious embodiments, power inverter 106 may operate using high-frequencytransformers, low-frequency transformers, or without transformers.Low-frequency transformers may convert the DC output from battery 108directly to the AC output provided to energy grid 104. High-frequencytransformers may employ a multi-step process that involves convertingthe DC output to high-frequency AC, then back to DC, and then finally tothe AC output provided to energy grid 104.

System 100 is shown to include a point of interconnection (POI) 110. POI110 is the point at which campus 102, energy grid 104, and powerinverter 106 are electrically connected. The power supplied to POI 110from power inverter 106 is shown as P_(sup). P_(sup) may be defined asP_(bat)+P_(loss), where P_(batt) is the battery power and P_(loss) isthe power loss in the battery system (e.g., losses in power inverter 106and/or battery 108). P_(sup) may be positive is power inverter 106 isproviding power to POI 110 or negative if power inverter 106 isreceiving power from POI 110. P_(campus) and P_(sup) combine at POI 110to form P_(POI). P_(POI) may be defined as the power provided to energygrid 104 from POI 110. P_(POI) may be positive if POI 110 is providingpower to energy grid 104 or negative if POI 110 is receiving power fromenergy grid 104.

Still referring to FIG. 1, system 100 is shown to include a frequencyresponse controller 112. Controller 112 may be configured to generateand provide power setpoints to power inverter 106. Power inverter 106may use the power setpoints to control the amount of power P_(sup)provided to POI 110 or drawn from POI 110. For example, power inverter106 may be configured to draw power from POI 110 and store the power inbattery 108 in response to receiving a negative power setpoint fromcontroller 112. Conversely, power inverter 106 may be configured to drawpower from battery 108 and provide the power to POI 110 in response toreceiving a positive power setpoint from controller 112. The magnitudeof the power setpoint may define the amount of power P_(sup) provided toor from power inverter 106. Controller 112 may be configured to generateand provide power setpoints that optimize the value of operating system100 over a time horizon.

In some embodiments, frequency response controller 112 uses powerinverter 106 and battery 108 to perform frequency regulation for energygrid 104. Frequency regulation is the process of maintaining thestability of the grid frequency (e.g., 60 Hz in the United States). Thegrid frequency may remain stable and balanced as long as the totalelectric supply and demand of energy grid 104 are balanced. Anydeviation from that balance may result in a deviation of the gridfrequency from its desirable value. For example, an increase in demandmay cause the grid frequency to decrease, whereas an increase in supplymay cause the grid frequency to increase. Frequency response controller112 may be configured to offset a fluctuation in the grid frequency bycausing power inverter 106 to supply energy from battery 108 to energygrid 104 (e.g., to offset a decrease in grid frequency) or store energyfrom energy grid 104 in battery 108 (e.g., to offset an increase in gridfrequency).

In some embodiments, frequency response controller 112 uses powerinverter 106 and battery 108 to perform load shifting for campus 102.For example, controller 112 may cause power inverter 106 to store energyin battery 108 when energy prices are low and retrieve energy frombattery 108 when energy prices are high in order to reduce the cost ofelectricity required to power campus 102. Load shifting may also allowsystem 100 reduce the demand charge incurred. Demand charge is anadditional charge imposed by some utility providers based on the maximumpower consumption during an applicable demand charge period. Forexample, a demand charge rate may be specified in terms of dollars perunit of power (e.g., $/kW) and may be multiplied by the peak power usage(e.g., kW) during a demand charge period to calculate the demand charge.Load shifting may allow system 100 to smooth momentary spikes in theelectric demand of campus 102 by drawing energy from battery 108 inorder to reduce peak power draw from energy grid 104, thereby decreasingthe demand charge incurred.

Still referring to FIG. 1, system 100 is shown to include an incentiveprovider 114. Incentive provider 114 may be a utility (e.g., an electricutility), a regional transmission organization (RTO), an independentsystem operator (ISO), or any other entity that provides incentives forperforming frequency regulation. For example, incentive provider 114 mayprovide system 100 with monetary incentives for participating in afrequency response program. In order to participate in the frequencyresponse program, system 100 may maintain a reserve capacity of storedenergy (e.g., in battery 108) that can be provided to energy grid 104.System 100 may also maintain the capacity to draw energy from energygrid 104 and store the energy in battery 108. Reserving both of thesecapacities may be accomplished by managing the state-of-charge ofbattery 108.

Frequency response controller 112 may provide incentive provider 114with a price bid and a capability bid. The price bid may include a priceper unit power (e.g., $/MW) for reserving or storing power that allowssystem 100 to participate in a frequency response program offered byincentive provider 114. The price per unit power bid by frequencyresponse controller 112 is referred to herein as the “capability price.”The price bid may also include a price for actual performance, referredto herein as the “performance price.” The capability bid may define anamount of power (e.g., MW) that system 100 will reserve or store inbattery 108 to perform frequency response, referred to herein as the“capability bid.”

Incentive provider 114 may provide frequency response controller 112with a capability clearing price CP_(cap), a performance clearing priceCP_(perf), and a regulation award Reg_(award), which correspond to thecapability price, the performance price, and the capability bid,respectively. In some embodiments, CP_(cap), CP_(perf), and Reg_(award)are the same as the corresponding bids placed by controller 112. Inother embodiments, CP_(cap), CP_(perf), and Reg_(award) may not be thesame as the bids placed by controller 112. For example, CP_(cap),CP_(perf), and Reg_(award) may be generated by incentive provider 114based on bids received from multiple participants in the frequencyresponse program. Controller 112 may use CP_(cap), CP_(perf), andReg_(award) to perform frequency regulation, described in greater detailbelow.

Frequency response controller 112 is shown receiving a regulation signalfrom incentive provider 114. The regulation signal may specify a portionof the regulation award Reg_(award) that frequency response controller112 is to add or remove from energy grid 104. In some embodiments, theregulation signal is a normalized signal (e.g., between −1 and 1)specifying a proportion of Reg_(award). Positive values of theregulation signal may indicate an amount of power to add to energy grid104, whereas negative values of the regulation signal may indicate anamount of power to remove from energy grid 104.

Frequency response controller 112 may respond to the regulation signalby generating an optimal power setpoint for power inverter 106. Theoptimal power setpoint may take into account both the potential revenuefrom participating in the frequency response program and the costs ofparticipation. Costs of participation may include, for example, amonetized cost of battery degradation as well as the energy and demandcharges that will be incurred. The optimization may be performed usingsequential quadratic programming, dynamic programming, or any otheroptimization technique.

In some embodiments, controller 112 uses a battery life model toquantify and monetize battery degradation as a function of the powersetpoints provided to power inverter 106. Advantageously, the batterylife model allows controller 112 to perform an optimization that weighsthe revenue generation potential of participating in the frequencyresponse program against the cost of battery degradation and other costsof participation (e.g., less battery power available for campus 102,increased electricity costs, etc.). An exemplary regulation signal andpower response are described in greater detail with reference to FIG. 2.

Referring now to FIG. 2, a pair of frequency response graphs 200 and 250are shown, according to an exemplary embodiment. Graph 200 illustrates aregulation signal Reg_(signal) 202 as a function of time. Reg_(signal)202 is shown as a normalized signal ranging from −1 to 1 (i.e.,−1≤Reg_(signal)≤1). Reg_(signal) 202 may be generated by incentiveprovider 114 and provided to frequency response controller 112.Reg_(signal) 202 may define a proportion of the regulation awardReg_(award) 254 that controller 112 is to add or remove from energy grid104, relative to a baseline value referred to as the midpoint b 256. Forexample, if the value of Reg_(award) 254 is 10 MW, a regulation signalvalue of 0.5 (i.e., Reg_(signal)=0.5) may indicate that system 100 isrequested to add 5 MW of power at POI 110 relative to midpoint b (e.g.,P*_(POI)=10 MW×0.5+b), whereas a regulation signal value of −0.3 mayindicate that system 100 is requested to remove 3 MW of power from POI110 relative to midpoint b (e.g., P*_(POI), =10 MW×−0.3+b).

Graph 250 illustrates the desired interconnection power P*_(POI) 252 asa function of time. P*_(POI) 252 may be calculated by frequency responsecontroller 112 based on Reg_(signal) 202, Reg_(award) 254, and amidpoint b 256. For example, controller 112 may calculate P*_(POI) 252using the following equation:P* _(POI)=Reg_(award)×Reg_(signal) +bwhere P*_(POI) represents the desired power at POI 110 (e.g.,P*_(POI)=P_(sup)+P_(campus)) and b is the midpoint. Midpoint b may bedefined (e.g., set or optimized) by controller 112 and may represent themidpoint of regulation around which the load is modified in response toReg_(signal) 202. Optimal adjustment of midpoint b may allow controller112 to actively participate in the frequency response market while alsotaking into account the energy and demand charge that will be incurred.

In order to participate in the frequency response market, controller 112may perform several tasks. Controller 112 may generate a price bid(e.g., $/MW) that includes the capability price and the performanceprice. In some embodiments, controller 112 sends the price bid toincentive provider 114 at approximately 15:30 each day and the price bidremains in effect for the entirety of the next day. Prior to beginning afrequency response period, controller 112 may generate the capabilitybid (e.g., MW) and send the capability bid to incentive provider 114. Insome embodiments, controller 112 generates and sends the capability bidto incentive provider 114 approximately 1.5 hours before a frequencyresponse period begins. In an exemplary embodiment, each frequencyresponse period has a duration of one hour; however, it is contemplatedthat frequency response periods may have any duration.

At the start of each frequency response period, controller 112 maygenerate the midpoint b around which controller 112 plans to performfrequency regulation. In some embodiments, controller 112 generates amidpoint b that will maintain battery 108 at a constant state-of-charge(SOC) (i.e. a midpoint that will result in battery 108 having the sameSOC at the beginning and end of the frequency response period). In otherembodiments, controller 112 generates midpoint b using an optimizationprocedure that allows the SOC of battery 108 to have different values atthe beginning and end of the frequency response period. For example,controller 112 may use the SOC of battery 108 as a constrained variablethat depends on midpoint b in order to optimize a value function thattakes into account frequency response revenue, energy costs, and thecost of battery degradation. Exemplary processes for calculating and/oroptimizing midpoint b under both the constant SOC scenario and thevariable SOC scenario are described in greater detail with reference toFIGS. 3-4.

During each frequency response period, controller 112 may periodicallygenerate a power setpoint for power inverter 106. For example,controller 112 may generate a power setpoint for each time step in thefrequency response period. In some embodiments, controller 112 generatesthe power setpoints using the equation:P* _(POI)=Reg_(award)×Reg_(signal) +bwhere P*_(POI)=P_(sup)+P_(campus). Positive values of P*_(POI) indicateenergy flow from POI 110 to energy grid 104. Positive values of P_(sup)and P_(campus) indicate energy flow to POI 110 from power inverter 106and campus 102, respectively. In other embodiments, controller 112generates the power setpoints using the equation:P* _(POI)=Reg_(award)×Res_(FR) +bwhere Res_(FR) is an optimal frequency response generated by optimizinga value function (described in greater detail below). Controller 112 maysubtract P_(campus) from P*_(POI) to generate the power setpoint forpower inverter 106 (i.e., P_(sup)=P*_(POI)−P_(campus)). The powersetpoint for power inverter 106 indicates the amount of power that powerinverter 106 is to add to POI 110 (if the power setpoint is positive) orremove from POI 110 (if the power setpoint is negative).Frequency Response Controller

Referring now to FIG. 3, a block diagram illustrating frequency responsecontroller 112 in greater detail is shown, according to an exemplaryembodiment. Frequency response controller 112 may be configured toperform an optimization process to generate values for the bid price,the capability bid, and the midpoint b. In some embodiments, frequencyresponse controller 112 generates values for the bids and the midpoint bperiodically using a predictive optimization scheme (e.g., once everyhalf hour, once per frequency response period, etc.). Controller 112 mayalso calculate and update power setpoints for power inverter 106periodically during each frequency response period (e.g., once every twoseconds).

In some embodiments, the interval at which controller 112 generatespower setpoints for power inverter 106 is significantly shorter than theinterval at which controller 112 generates the bids and the midpoint b.For example, controller 112 may generate values for the bids and themidpoint b every half hour, whereas controller 112 may generate a powersetpoint for power inverter 106 every two seconds. The difference inthese time scales allows controller 112 to use a cascaded optimizationprocess to generate optimal bids, midpoints b, and power setpoints.

In the cascaded optimization process, a high level controller 312determines optimal values for the bid price, the capability bid, and themidpoint b by performing a high level optimization. High levelcontroller 312 may select midpoint b to maintain a constantstate-of-charge in battery 108 (i.e., the same state-of-charge at thebeginning and end of each frequency response period) or to vary thestate-of-charge in order to optimize the overall value of operatingsystem 100 (e.g., frequency response revenue minus energy costs andbattery degradation costs). High level controller 312 may also determinefilter parameters for a signal filter (e.g., a low pass filter) used bya low level controller 314.

Low level controller 314 uses the midpoint b and the filter parametersfrom high level controller 312 to perform a low level optimization inorder to generate the power setpoints for power inverter 106.Advantageously, low level controller 314 may determine how closely totrack the desired power P*_(POI) at the point of interconnection 110.For example, the low level optimization performed by low levelcontroller 314 may consider not only frequency response revenue but alsothe costs of the power setpoints in terms of energy costs and batterydegradation. In some instances, low level controller 314 may determinethat it is deleterious to battery 108 to follow the regulation exactlyand may sacrifice a portion of the frequency response revenue in orderto preserve the life of battery 108. The cascaded optimization processis described in greater detail below.

Still referring to FIG. 3, frequency response controller 112 is shown toinclude a communications interface 302 and a processing circuit 304.Communications interface 302 may include wired or wireless interfaces(e.g., jacks, antennas, transmitters, receivers, transceivers, wireterminals, etc.) for conducting data communications with varioussystems, devices, or networks. For example, communications interface 302may include an Ethernet card and port for sending and receiving data viaan Ethernet-based communications network and/or a WiFi transceiver forcommunicating via a wireless communications network. Communicationsinterface 302 may be configured to communicate via local area networksor wide area networks (e.g., the Internet, a building WAN, etc.) and mayuse a variety of communications protocols (e.g., BACnet, IP, LON, etc.).

Communications interface 302 may be a network interface configured tofacilitate electronic data communications between frequency responsecontroller 112 and various external systems or devices (e.g., campus102, energy grid 104, power inverter 106, incentive provider 114,utilities 320, weather service 322, etc.). For example, frequencyresponse controller 112 may receive inputs from incentive provider 114indicating an incentive event history (e.g., past clearing prices,mileage ratios, participation requirements, etc.) and a regulationsignal. Controller 112 may receive a campus power signal from campus102, utility rates from utilities 320, and weather forecasts fromweather service 322 via communications interface 302. Controller 112 mayprovide a price bid and a capability bid to incentive provider 114 andmay provide power setpoints to power inverter 106 via communicationsinterface 302.

Still referring to FIG. 3, processing circuit 304 is shown to include aprocessor 306 and memory 308. Processor 306 may be a general purpose orspecific purpose processor, an application specific integrated circuit(ASIC), one or more field programmable gate arrays (FPGAs), a group ofprocessing components, or other suitable processing components.Processor 306 may be configured to execute computer code or instructionsstored in memory 308 or received from other computer readable media(e.g., CDROM, network storage, a remote server, etc.).

Memory 308 may include one or more devices (e.g., memory units, memorydevices, storage devices, etc.) for storing data and/or computer codefor completing and/or facilitating the various processes described inthe present disclosure. Memory 308 may include random access memory(RAM), read-only memory (ROM), hard drive storage, temporary storage,non-volatile memory, flash memory, optical memory, or any other suitablememory for storing software objects and/or computer instructions. Memory308 may include database components, object code components, scriptcomponents, or any other type of information structure for supportingthe various activities and information structures described in thepresent disclosure. Memory 308 may be communicably connected toprocessor 306 via processing circuit 304 and may include computer codefor executing (e.g., by processor 306) one or more processes describedherein.

Still referring to FIG. 3, frequency response controller 112 is shown toinclude a load/rate predictor 310. Load/rate predictor 310 may beconfigured to predict the electric load of campus 102 (i.e., {circumflexover (P)}_(campus)) for each time step k (e.g., k=1 . . . n) within anoptimization window. Load/rate predictor 310 is shown receiving weatherforecasts from a weather service 322. In some embodiments, load/ratepredictor 310 predicts {circumflex over (P)}_(campus) as a function ofthe weather forecasts. In some embodiments, load/rate predictor 310 usesfeedback from campus 102 to predict {circumflex over (P)}_(campus).Feedback from campus 102 may include various types of sensory inputs(e.g., temperature, flow, humidity, enthalpy, etc.) or other datarelating to buildings 116, central plant 118, and/or energy generation120 (e.g., inputs from a HVAC system, a lighting control system, asecurity system, a water system, a PV energy system, etc.). Load/ratepredictor 310 may predict one or more different types of loads forcampus 102. For example, load/rate predictor 310 may predict a hot waterload, a cold water load, and/or an electric load for each time step kwithin the optimization window.

In some embodiments, load/rate predictor 310 receives a measuredelectric load and/or previous measured load data from campus 102. Forexample, load/rate predictor 310 is shown receiving a campus powersignal from campus 102. The campus power signal may indicate themeasured electric load of campus 102. Load/rate predictor 310 maypredict one or more statistics of the campus power signal including, forexample, a mean campus power μ_(campus) and a standard deviation of thecampus power σ_(campus). Load/rate predictor 310 may predict {circumflexover (P)}_(campus) as a function of a given weather forecast({circumflex over (ϕ)}_(w)), a day type (day), the time of day (t), andprevious measured load data (Y_(k−1)). Such a relationship is expressedin the following equation:{circumflex over (P)} _(campus) =f({circumflex over (ϕ)}_(w),day,t|Y_(k−1))

In some embodiments, load/rate predictor 310 uses a deterministic plusstochastic model trained from historical load data to predict{circumflex over (P)}_(campus). Load/rate predictor 310 may use any of avariety of prediction methods to predict {circumflex over (P)}_(campus)(e.g., linear regression for the deterministic portion and an AR modelfor the stochastic portion). In some embodiments, load/rate predictor310 makes load/rate predictions using the techniques described in U.S.patent application Ser. No. 14/717,593, titled “Building ManagementSystem for Forecasting Time Series Values of Building Variables” andfiled May 20, 2015.

Load/rate predictor 310 is shown receiving utility rates from utilities320. Utility rates may indicate a cost or price per unit of a resource(e.g., electricity, natural gas, water, etc.) provided by utilities 320at each time step k in the optimization window. In some embodiments, theutility rates are time-variable rates. For example, the price ofelectricity may be higher at certain times of day or days of the week(e.g., during high demand periods) and lower at other times of day ordays of the week (e.g., during low demand periods). The utility ratesmay define various time periods and a cost per unit of a resource duringeach time period. Utility rates may be actual rates received fromutilities 320 or predicted utility rates estimated by load/ratepredictor 310.

In some embodiments, the utility rates include demand charges for one ormore resources provided by utilities 320. A demand charge may define aseparate cost imposed by utilities 320 based on the maximum usage of aparticular resource (e.g., maximum energy consumption) during a demandcharge period. The utility rates may define various demand chargeperiods and one or more demand charges associated with each demandcharge period. In some instances, demand charge periods may overlappartially or completely with each other and/or with the predictionwindow. Advantageously, frequency response controller 112 may beconfigured to account for demand charges in the high level optimizationprocess performed by high level controller 312. Utilities 320 may bedefined by time-variable (e.g., hourly) prices, a maximum service level(e.g., a maximum rate of consumption allowed by the physicalinfrastructure or by contract) and, in the case of electricity, a demandcharge or a charge for the peak rate of consumption within a certainperiod. Load/rate predictor 310 may store the predicted campus power{circumflex over (P)}_(campus) and the utility rates in memory 308and/or provide the predicted campus power {circumflex over (P)}_(campus)and the utility rates to high level controller 312.

Still referring to FIG. 3, frequency response controller 112 is shown toinclude an energy market predictor 316 and a signal statistics predictor318. Energy market predictor 316 may be configured to predict energymarket statistics relating to the frequency response program. Forexample, energy market predictor 316 may predict the values of one ormore variables that can be used to estimate frequency response revenue.In some embodiments, the frequency response revenue is defined by thefollowing equation:Rev=PS(CP_(cap)+MR+CP_(perf))Reg_(award)where Rev is the frequency response revenue, CP_(cap) is the capabilityclearing price, MR is the mileage ratio, and CP_(perf) is theperformance clearing price. PS is a performance score based on howclosely the frequency response provided by controller 112 tracks theregulation signal. Energy market predictor 316 may be configured topredict the capability clearing price CP_(cap), the performance clearingprice CP_(perf), the mileage ratio MR, and/or other energy marketstatistics that can be used to estimate frequency response revenue.Energy market predictor 316 may store the energy market statistics inmemory 308 and/or provide the energy market statistics to high levelcontroller 312.

Signal statistics predictor 318 may be configured to predict one or morestatistics of the regulation signal provided by incentive provider 114.For example, signal statistics predictor 318 may be configured topredict the mean μ_(FR), standard deviation σ_(FR), and/or otherstatistics of the regulation signal. The regulation signal statisticsmay be based on previous values of the regulation signal (e.g., ahistorical mean, a historical standard deviation, etc.) or predictedvalues of the regulation signal (e.g., a predicted mean, a predictedstandard deviation, etc.).

In some embodiments, signal statistics predictor 318 uses adeterministic plus stochastic model trained from historical regulationsignal data to predict future values of the regulation signal. Forexample, signal statistics predictor 318 may use linear regression topredict a deterministic portion of the regulation signal and an AR modelto predict a stochastic portion of the regulation signal. In someembodiments, signal statistics predictor 318 predicts the regulationsignal using the techniques described in U.S. patent application Ser.No. 14/717,593, titled “Building Management System for Forecasting TimeSeries Values of Building Variables” and filed May 20, 2015. Signalstatistics predictor 318 may use the predicted values of the regulationsignal to calculate the regulation signal statistics. Signal statisticspredictor 318 may store the regulation signal statistics in memory 308and/or provide the regulation signal statistics to high level controller312.

Still referring to FIG. 3, frequency response controller 112 is shown toinclude a high level controller 312. High level controller 312 may beconfigured to generate values for the midpoint b and the capability bidReg_(award). In some embodiments, high level controller 312 determines amidpoint b that will cause battery 108 to have the same state-of-charge(SOC) at the beginning and end of each frequency response period. Inother embodiments, high level controller 312 performs an optimizationprocess to generate midpoint b and Reg_(award). For example, high levelcontroller 312 may generate midpoint b using an optimization procedurethat allows the SOC of battery 108 to vary and/or have different valuesat the beginning and end of the frequency response period. High levelcontroller 312 may use the SOC of battery 108 as a constrained variablethat depends on midpoint b in order to optimize a value function thattakes into account frequency response revenue, energy costs, and thecost of battery degradation. Both of these embodiments are described ingreater detail with reference to FIG. 4.

High level controller 312 may determine midpoint b by equating thedesired power P*_(POI) at POI 110 with the actual power at POI 110 asshown in the following equation:(Reg_(signal))(Reg_(award))+b=P _(bat) +P _(loss) +P _(campus)where the left side of the equation (Reg_(signal))(Reg_(award))+b is thedesired power P*_(POI) at POI 110 and the right side of the equation isthe actual power at POI 110. Integrating over the frequency responseperiod results in the following equation:

∫_(period)((Reg_(signal))(Reg_(award)) + b)dt= ∫_(period)(P_(bat) + P_(loss) + P_(campus))dt

For embodiments in which the SOC of battery 108 is maintained at thesame value at the beginning and end of the frequency response period,the integral of the battery power P_(bat) over the frequency responseperiod is zero (i.e., ∫P_(bat)dt=0). Accordingly, the previous equationcan be rewritten as follows:

b = ∫_(period)P_(loss)dt + ∫_(period)P_(campus)dt − Reg_(award)∫_(period)Reg_(signal)dtwhere the term ∫P_(bat)dt has been omitted because ∫P_(bat)dt=0. This isideal behavior if the only goal is to maximize frequency responserevenue. Keeping the SOC of battery 108 at a constant value (and near50%) will allow system 100 to participate in the frequency market duringall hours of the day.

High level controller 312 may use the estimated values of the campuspower signal received from campus 102 to predict the value of∫P_(campus)dt over the frequency response period. Similarly, high levelcontroller 312 may use the estimated values of the regulation signalfrom incentive provider 114 to predict the value of ∫Reg_(signal)dt overthe frequency response period. High level controller 312 may estimatethe value of ∫P_(loss)dt using a Thevinin equivalent circuit model ofbattery 108 (described in greater detail with reference to FIG. 4). Thisallows high level controller 312 to estimate the integral ∫P_(loss)dt asa function of other variables such as Reg_(award), Reg_(signal),P_(campus), and midpoint b.

After substituting known and estimated values, the preceding equationcan be rewritten as follows:

$\frac{1}{4P_{\max}}\left\lbrack {{E\left\{ P_{campus}^{2} \right\}} + {{Reg}_{award}^{2}E\left\{ {Reg}_{signal}^{2} \right\}} - {\left. \quad{2{Reg}_{award}E\left\{ {Reg}_{signal} \right\} E\left\{ P_{campus} \right\}} \right\rbrack\Delta\; t} + {\quad{{{\left\lbrack {{{Reg}_{award}E\left\{ {Reg}_{signal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack\Delta\; t} + {{\frac{b}{2P_{\max}}\left\lbrack {{{Reg}_{award}E\left\{ {Reg}_{signal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} + {b\;\Delta\; t} + {\frac{b^{2}}{4P_{\max}}\Delta\; t}} = 0}}} \right.$where the notation E{ } indicates that the variable within the brackets{ } is ergodic and can be approximated by the estimated mean of thevariable. For example, the term E{Reg_(signal)} can be approximated bythe estimated mean of the regulation signal μ_(FR) and the termE{P_(campus)} can be approximated by the estimated mean of the campuspower signal μ_(campus). High level controller 312 may solve theequation for midpoint b to determine the midpoint b that maintainsbattery 108 at a constant state-of-charge.

For embodiments in which the SOC of battery 108 is treated as avariable, the SOC of battery 108 may be allowed to have different valuesat the beginning and end of the frequency response period. Accordingly,the integral of the battery power P_(bat) over the frequency responseperiod can be expressed as −ΔSOC·C_(des) as shown in the followingequation:

 ∫_(period)P_(bat)dt = −Δ SOC ⋅ C_(des)where ΔSOC is the change in the SOC of battery 108 over the frequencyresponse period and C_(des) is the design capacity of battery 108. TheSOC of battery 108 may be a normalized variable (i.e., 0≤SOC≤1) suchthat the term SOC·C_(des) represents the amount of energy stored inbattery 108 for a given state-of-charge. The SOC is shown as a negativevalue because drawing energy from battery 108 (i.e., a positive P_(bat))decreases the SOC of battery 108. The equation for midpoint b becomes:

$b = {{\int\limits_{period}{P_{loss}\mspace{11mu}{dt}}} + {\int\limits_{period}{P_{campus}\mspace{11mu}{dt}}} + {\int\limits_{period}{P_{bat}\mspace{11mu}{dt}}} - {{Reg}_{award}{\int\limits_{period}{{Reg}_{signal}\mspace{11mu}{dt}}}}}$

After substituting known and estimated values, the preceding equationcan be rewritten as follows:

${{{{\frac{1}{4P_{\max}}\left\lbrack {{E\left\{ P_{campus}^{2} \right\}} + {Reg}_{award}^{2} + {E\left\{ {Reg}_{signal}^{2} \right\}} - {2{Reg}_{award}E\left\{ {Reg}_{signal} \right\} E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} +}\quad}{\quad{{{\left\lbrack {{{Reg}_{award}E\left\{ {Reg}_{signal} \right\}} + {E\left\{ P_{campus} \right\}}} \right\rbrack\Delta\; t} + {\Delta\;{{SOC} \cdot C_{des}}} + {{\frac{b}{2P_{\max}}\left\lbrack {{{Reg}_{award}E\left\{ {Reg}_{signal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} + {b\;\Delta\; t} + {\frac{b^{2}}{4P_{\max}}\Delta\; t}} = 0}}$High level controller 312 may solve the equation for midpoint b in termsof ΔSOC.

High level controller 312 may perform an optimization to find optimalmidpoints b for each frequency response period within an optimizationwindow (e.g., each hour for the next day) given the electrical costsover the optimization window. Optimal midpoints b may be the midpointsthat maximize an objective function that includes both frequencyresponse revenue and costs of electricity and battery degradation. Forexample, an objective function J can be written as:

$J = {{\sum\limits_{k = 1}^{h}{{Rev}\left( {Reg}_{{award},k} \right)}} + {\sum\limits_{k = 1}^{h}{c_{k}b_{k}}} + {\min\limits_{period}\left( {P_{{campus},k} + b_{k}} \right)} - {\sum\limits_{k = 1}^{h}\lambda_{{bat},k}}}$where Rev(Reg_(award,k)) is the frequency response revenue at time stepk, c_(k)b_(k) is the cost of electricity purchased at time step k, themin( ) term is the demand charge based on the maximum rate ofelectricity consumption during the applicable demand charge period, andλ_(bat,k) is the monetized cost battery degradation at time step k. Theelectricity cost is expressed as a positive value because drawing powerfrom energy grid 104 is represented as a negative number and thereforewill result in negative value (i.e., a cost) in the objective function.The demand charge is expressed as a minimum for the same reason (i.e.,the most negative power value represents maximum power draw from energygrid 104).

High level controller 312 may estimate the frequency response revenueRev(Reg_(award,k)) as a function of the midpoints b. In someembodiments, high level controller 312 estimates frequency responserevenue using the following equation:Rev(Reg_(award))=Reg_(award)(CP_(cap)+MR+CP_(perf))where CP_(cap), MR, and CP_(perf) are the energy market statisticsreceived from energy market predictor 316 and Reg_(award) is a functionof the midpoint b. For example, high level controller 312 may place abid that is as large as possible for a given midpoint, as shown in thefollowing equation:Reg_(award) =P _(limit) −|b|where P_(limit) is the power rating of power inverter 106.Advantageously, selecting Reg_(award) as a function of midpoint b allowshigh level controller 312 to predict the frequency response revenue thatwill result from a given midpoint b.

High level controller 312 may estimate the cost of battery degradationλ_(bat) as a function of the midpoints b. For example, high levelcontroller 312 may use a battery life model to predict a loss in batterycapacity that will result from a set of midpoints b, power outputs,and/or other variables that can be manipulated by controller 112. Insome embodiments, the battery life model expresses the loss in batterycapacity C_(loss,add) as a sum of multiple piecewise linear functions,as shown in the following equation:C _(loss,add) =f ₁(T _(cell))+f ₂(SOC)+f ₃(DOD)+f ₄(PR)+f ₅(ER)−C_(loss,nom)where T_(cell) is the cell temperature, SOC is the state-of-charge, DODis the depth of discharge, PR is the average power ratio

$\left( {{e.g.},{{PR} = {{avg}\mspace{11mu}\left( \frac{P}{P_{des}} \right)}}} \right),$and ER is the average effort ratio

$\left( {{e.g.},{{ER} = {{avg}\mspace{11mu}\left( \frac{\Delta\; P}{P_{des}} \right)}}} \right.$of battery 108. Each of these terms is described in greater detail withreference to FIG. 4. Advantageously, several of the terms in the batterylife model depend on the midpoints b and power setpoints selected bycontroller 112. This allows high level controller 312 to predict a lossin battery capacity that will result from a given set of controloutputs. High level controller 312 may monetize the loss in batterycapacity and include the monetized cost of battery degradation λ_(bat)in the objective function J.

In some embodiments, high level controller 312 generates a set of filterparameters for low level controller 314. The filter parameters may beused by low level controller 314 as part of a low-pass filter thatremoves high frequency components from the regulation signal. In someembodiments, high level controller 312 generates a set of filterparameters that transform the regulation signal into an optimalfrequency response signal Res_(FR). For example, high level controller312 may perform a second optimization process to determine an optimalfrequency response Res_(FR) based on the optimized values forReg_(award) and midpoint b.

In some embodiments, high level controller 312 determines the optimalfrequency response Res_(FR) by optimizing value function J with thefrequency response revenue Rev(Reg_(award)) defined as follows:Rev(Reg_(award))=PS·Reg_(award)(CP_(cap)+MR·CP_(perf))and with the frequency response Res_(FR) substituted for the regulationsignal in the battery life model. The performance score PS may be basedon several factors that indicate how well the optimal frequency responseRes_(FR) tracks the regulation signal. Closely tracking the regulationsignal may result in higher performance scores, thereby increasing thefrequency response revenue. However, closely tracking the regulationsignal may also increase the cost of battery degradation λ_(bat). Theoptimized frequency response Res_(FR) represents an optimal tradeoffbetween decreased frequency response revenue and increased battery life.High level controller 312 may use the optimized frequency responseRes_(FR) to generate a set of filter parameters for low level controller314. These and other features of high level controller 312 are describedin greater detail with reference to FIG. 4.

Still referring to FIG. 3, frequency response controller 112 is shown toinclude a low level controller 314. Low level controller 314 is shownreceiving the midpoints b and the filter parameters from high levelcontroller 312. Low level controller 314 may also receive the campuspower signal from campus 102 and the regulation signal from incentiveprovider 114.

Low level controller 314 may use the regulation signal to predict futurevalues of the regulation signal and may filter the predicted regulationsignal using the filter parameters provided by high level controller312.

Low level controller 314 may use the filtered regulation signal todetermine optimal power setpoints for power inverter 106. For example,low level controller 314 may use the filtered regulation signal tocalculate the desired interconnection power P*_(POI) using the followingequation:P* _(POI)=Reg_(award)·Reg_(filter) +bwhere Reg_(filter) is the filtered regulation signal. Low levelcontroller 314 may subtract the campus power P_(campus) from the desiredinterconnection power P*_(POI) to calculate the optimal power setpointsP_(SP) for power inverter 106, as shown in the following equation:P _(SP) =P _(POI) −P _(campus)

In some embodiments, low level controller 314 performs an optimizationto determine how closely to track P*_(POI). For example, low levelcontroller 314 may determine an optimal frequency response Res_(FR) byoptimizing value function J with the frequency response revenueRev(Reg_(award)) defined as follows:Rev(Reg_(award))=PS·Reg_(award)(CP_(cap)+MR·CP_(perf))and with the frequency response Res_(FR) substituted for the regulationsignal in the battery life model. Low level controller 314 may use theoptimal frequency response Res_(FR) in place of the filtered frequencyresponse Reg_(filter) to calculate the desired interconnection powerP*_(POI) and power setpoints P_(SP) as previously described. These andother features of low level controller 314 are described in greaterdetail with reference to FIG. 5.High Level Controller

Referring now to FIG. 4, a block diagram illustrating high levelcontroller 312 in greater detail is shown, according to an exemplaryembodiment. High level controller 312 is shown to include a constantstate-of-charge (SOC) controller 402 and a variable SOC controller 408.Constant SOC controller 402 may be configured to generate a midpoint bthat results in battery 108 having the same SOC at the beginning and theend of each frequency response period. In other words, constant SOCcontroller 408 may determine a midpoint b that maintains battery 108 ata predetermined SOC at the beginning of each frequency response period.Variable SOC controller 408 may generate midpoint b using anoptimization procedure that allows the SOC of battery 108 to havedifferent values at the beginning and end of the frequency responseperiod. In other words, variable SOC controller 408 may determine amidpoint b that results in a net change in the SOC of battery 108 overthe duration of the frequency response period.

Constant State-of-Charge Controller

Constant SOC controller 402 may determine midpoint b by equating thedesired power P*_(POI) at POI 110 with the actual power at POI 110 asshown in the following equation:(Reg_(signal))(Reg_(award))+b=P _(bat) +P _(loss) +P _(campus)where the left side of the equation (Reg_(signal))(Reg_(award))+b is thedesired power P*_(POI) at POI 110 and the right side of the equation isthe actual power at POI 110. Integrating over the frequency responseperiod results in the following equation:

${\int\limits_{period}{\left( {{\left( {Reg}_{signal} \right)\left( {Reg}_{award} \right)} + b} \right)\mspace{11mu}{dt}}} = {\int\limits_{period}{\left( {P_{bat} + P_{loss} + P_{campus}} \right)\mspace{11mu}{dt}}}$

Since the SOC of battery 108 is maintained at the same value at thebeginning and end of the frequency response period, the integral of thebattery power P_(bat) over the frequency response period is zero (i.e.,∫P_(bat)dt=0). Accordingly, the previous equation can be rewritten asfollows:

$b = {{\int\limits_{period}{P_{loss}\mspace{11mu}{dt}}} + {\int\limits_{period}{P_{campus}\mspace{11mu}{dt}}} - {{Reg}_{award}{\int\limits_{period}{{Reg}_{signal}\mspace{11mu}{dt}}}}}$where the term ∫P_(bat)dt has been omitted because ∫P_(bat)dt=0. This isideal behavior if the only goal is to maximize frequency responserevenue. Keeping the SOC of battery 108 at a constant value (and near50%) will allow system 100 to participate in the frequency market duringall hours of the day.

Constant SOC controller 402 may use the estimated values of the campuspower signal received from campus 102 to predict the value of∫P_(campus)dt over the frequency response period. Similarly, constantSOC controller 402 may use the estimated values of the regulation signalfrom incentive provider 114 to predict the value of ∫Reg_(signal)dt overthe frequency response period. Reg_(award) can be expressed as afunction of midpoint b as previously described (e.g.,Reg_(award)=P_(limit)−|b|). Therefore, the only remaining term in theequation for midpoint b is the expected battery power loss ∫P_(loss).

Constant SOC controller 402 is shown to include a battery power lossestimator 404. Battery power loss estimator 404 may estimate the valueof ∫P_(loss)dt using a Thevinin equivalent circuit model of battery 108.For example, battery power loss estimator 404 may model battery 108 as avoltage source in series with a resistor. The voltage source has an opencircuit voltage of V_(OC) and the resistor has a resistance of R_(TH).An electric current I flows from the voltage source through theresistor.

To find the battery power loss in terms of the supplied power P_(sup),battery power loss estimator 404 may identify the supplied power P_(sup)as a function of the current I, the open circuit voltage V_(OC), and theresistance R_(TH) as shown in the following equation:P _(sup) =V _(oc) I−I ² R _(TH)which can be rewritten as:

${\frac{I^{2}}{I_{SC}} - I + \frac{P^{\;\prime}}{4}} = 0$with the following substitutions:

$\begin{matrix}{{I_{SC} = \frac{V_{OC}}{R_{TH}}},} & {{P^{\;\prime} = \frac{P}{P_{\max}}},} & {P_{\max} = \frac{V_{OC}^{2}}{4R_{TH}}}\end{matrix}$where P is the supplied power and P_(max) is the maximum possible powertransfer.

Battery power loss estimator 404 may solve for the current I as follows:

$I = {\frac{I_{SC}}{2}\left( {1 - \sqrt{1 - P^{\;\prime}}} \right)}$which can be converted into an expression for power loss P_(loss) interms of the supplied power P and the maximum possible power transferP_(max) as shown in the following equation:P _(loss) =P _(max)(1−√{square root over (1−P′)})²

Battery power loss estimator 404 may simplify the previous equation byapproximating the expression (1−√{square root over (1−P′)}) as a linearfunction about P′=0. This results in the following approximation forP_(loss):

$P_{loss} \approx {P_{\max}\left( \frac{P^{\;\prime}}{2} \right)}^{2}$which is a good approximation for powers up to one-fifth of the maximumpower.

Battery power loss estimator 404 may calculate the expected value of∫P_(loss)dt over the frequency response period as follows:

$\quad\begin{matrix}{{\int\limits_{period}{P_{loss}\mspace{11mu}{dt}}} = {\int\limits_{period}{{- {P_{\max}\left( \frac{{{Reg}_{award}{Reg}_{signal}} + b - P_{campus}}{2P_{\max}} \right)}^{2}}{dt}}}} \\{= {\frac{1}{4P_{\max}}\left\lbrack {{2{Reg}_{award}{\int\limits_{period}{P_{campus}{Reg}_{signal}\mspace{11mu}{dt}}}} -} \right.}} \\{\left. {{\int\limits_{period}{P_{campus}^{2}\mspace{11mu}{dt}}} - {{Reg}_{award}^{2}{\int\limits_{period}{{Reg}_{signal}^{2}\mspace{11mu}{dt}}}}} \right\rbrack +} \\{{\frac{b}{2P_{\max}}\left\lbrack {{\int\limits_{period}{P_{campus}^{2}\mspace{11mu}{dt}}} - {{Reg}_{award}{\int\limits_{period}{{Reg}_{signal}\mspace{11mu}{dt}}}}} \right\rbrack} -} \\{\frac{b^{2}}{4P_{\max}}\Delta\; t} \\{= {\frac{1}{4P_{\max}}\left\lbrack {{E\left\{ P_{campus}^{2} \right\}} + {{Reg}_{award}^{2}E\left\{ {Reg}_{signal}^{2} \right\}} -} \right.}} \\{{\left. {2{Reg}_{award}\; E\left\{ {Reg}_{signal} \right\} E\left\{ P_{campus} \right\}} \right\rbrack\Delta\; t} -} \\{{{\frac{b}{2P_{\max}}\left\lbrack {{{Reg}_{award}\; E\left\{ {Reg}_{signal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} -} \\{\frac{b^{2}}{4P_{\max}}\Delta\; t}\end{matrix}$where the notation E{ } indicates that the variable within the brackets{ } is ergodic and can be approximated by the estimated mean of thevariable. This formulation allows battery power loss estimator 404 toestimate ∫P_(loss)dt as a function of other variables such asReg_(award), Reg_(signal), P_(campus), midpoint b, and P_(max).

Constant SOC controller 402 is shown to include a midpoint calculator406. Midpoint calculator 406 may be configured to calculate midpoint bby substituting the previous expression for ∫P_(loss)dt into theequation for midpoint b. After substituting known and estimated values,the equation for midpoint b can be rewritten as follows:

${{{{\frac{1}{4P_{\max}}\left\lbrack {{E\left\{ P_{campus}^{2} \right\}} + {{Reg}_{award}^{2}\; E\left\{ {Reg}_{signal}^{2} \right\}} - {2{Reg}_{award}\; E\left\{ {Reg}_{signal} \right\} E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} +}\quad}{\quad{{{\left\lbrack {{{Reg}_{award}\; E\left\{ {Reg}_{signal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack\Delta\; t} + {{\frac{b}{2P_{\max}}\left\lbrack {{{Reg}_{award}\; E\left\{ {Reg}_{signal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} + {b\;\Delta\; t} + {\frac{b^{2}}{4P_{\max}}\Delta\; t}} = 0}}$Midpoint calculator 406 may solve the equation for midpoint b todetermine the midpoint b that maintains battery 108 at a constantstate-of-charge.Variable State-of-Charge Controller

Variable SOC controller 408 may determine optimal midpoints b byallowing the SOC of battery 108 to have different values at thebeginning and end of a frequency response period. For embodiments inwhich the SOC of battery 108 is allowed to vary, the integral of thebattery power P_(bat) over the frequency response period can beexpressed as −ΔSOC·C_(des) as shown in the following equation:

${\int\limits_{period}{P_{bat}\mspace{11mu}{dt}}} = {{- \Delta}\;{{SOC} \cdot C_{des}}}$where ΔSOC is the change in the SOC of battery 108 over the frequencyresponse period and C_(des) is the design capacity of battery 108. TheSOC of battery 108 may be a normalized variable (i.e., 0≤SOC≤1) suchthat the term SOC·C_(des) represents the amount of energy stored inbattery 108 for a given state-of-charge. The SOC is shown as a negativevalue because drawing energy from battery 108 (i.e., a positive P_(bat))decreases the SOC of battery 108. The equation for midpoint b becomes:

$b = {{\int\limits_{period}{P_{loss}\mspace{11mu}{dt}}} + {\int\limits_{period}{P_{campus}\mspace{11mu}{dt}}} + {\int\limits_{period}{P_{bat}\mspace{11mu}{dt}}} - {{Reg}_{award}{\int\limits_{period}{{Reg}_{signal}\mspace{11mu}{dt}}}}}$

Variable SOC controller 408 is shown to include a battery power lossestimator 410 and a midpoint optimizer 412. Battery power loss estimator410 may be the same or similar to battery power loss estimator 404.Midpoint optimizer 412 may be configured to establish a relationshipbetween the midpoint b and the SOC of battery 108. For example, aftersubstituting known and estimated values, the equation for midpoint b canbe written as follows:

${{{{\frac{1}{4P_{\max}}\left\lbrack {{E\left\{ P_{campus}^{2} \right\}} + {Reg}_{award}^{2}\; + {E\left\{ {Reg}_{signal}^{2} \right\}} - {2{Reg}_{award}\; E\left\{ {Reg}_{signal} \right\} E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} +}\quad}{\quad{{{\left\lbrack {{{Reg}_{award}\; E\left\{ {Reg}_{signal} \right\}} + {E\left\{ P_{campus} \right\}}} \right\rbrack\Delta\; t} + {\Delta\;{{SOC} \cdot C_{des}}} + {{\frac{b}{2P_{\max}}\left\lbrack {{{Reg}_{award}\; E\left\{ {Reg}_{signal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} + {b\;\Delta\; t} + {\frac{b^{2}}{4P_{\max}}\Delta\; t}} = 0}}$

Advantageously, the previous equation defines a relationship betweenmidpoint b and the change in SOC of battery 108. Midpoint optimizer 412may use this equation to determine the impact that different values ofmidpoint b have on the SOC in order to determine optimal midpoints b.This equation can also be used by midpoint optimizer 412 duringoptimization to translate constraints on the SOC in terms of midpoint b.For example, the SOC of battery 108 may be constrained between zero and1 (e.g., 0≤SOC≤1) since battery 108 cannot be charged in excess of itsmaximum capacity or depleted below zero. Midpoint optimizer 412 may usethe relationship between ΔSOC and midpoint b to constrain theoptimization of midpoint b to midpoint values that satisfy the capacityconstraint.

Midpoint optimizer 412 may perform an optimization to find optimalmidpoints b for each frequency response period within the optimizationwindow (e.g., each hour for the next day) given the electrical costsover the optimization window. Optimal midpoints b may be the midpointsthat maximize an objective function that includes both frequencyresponse revenue and costs of electricity and battery degradation. Forexample, an objective function J can be written as:

$J = {{\sum\limits_{k = 1}^{h}{{Rev}\left( {Reg}_{{award},k} \right)}} + {\sum\limits_{k = 1}^{h}{c_{k}b_{k}}} + {\min\limits_{period}\left( {P_{{campus},k} + b_{k}} \right)} - {\sum\limits_{k = 1}^{h}\lambda_{{bat},k}}}$where Rev(Reg_(award,k)) is the frequency response revenue at time stepk, c_(k)b_(k) is the cost of electricity purchased at time step k, themin( ) term is the demand charge based on the maximum rate ofelectricity consumption during the applicable demand charge period, andλ_(bat,k) is the monetized cost battery degradation at time step k.Midpoint optimizer 412 may use input from frequency response revenueestimator 416 (e.g., a revenue model) to determine a relationshipbetween midpoint b and Rev(Reg_(award,k)). Similarly, midpoint optimizer412 may use input from battery degradation estimator 418 and/or revenueloss estimator 420 to determine a relationship between midpoint b andthe monetized cost of battery degradation λ_(bat,k).

Still referring to FIG. 4, variable SOC controller 408 is shown toinclude an optimization constraints module 414. Optimization constraintsmodule 414 may provide one or more constraints on the optimizationperformed by midpoint optimizer 412. The optimization constraints may bespecified in terms of midpoint b or other variables that are related tomidpoint b. For example, optimization constraints module 414 mayimplement an optimization constraint specifying that the expected SOC ofbattery 108 at the end of each frequency response period is between zeroand one, as shown in the following equation:

${0 \leq {{SOC}_{0} + {\sum\limits_{i = 1}^{j}{\Delta\;{SOC}_{i}}}} \leq {1\mspace{14mu}{\forall j}}} = {1\mspace{14mu}\ldots\mspace{14mu} h}$where SOC₀ is the SOC of battery 108 at the beginning of theoptimization window, ΔSOC_(i) is the change in SOC during frequencyresponse period i, and h is the total number of frequency responseperiods within the optimization window.

In some embodiments, optimization constraints module 414 implements anoptimization constraint on midpoint b so that the power at POI 110 doesnot exceed the power rating of power inverter 106. Such a constraint isshown in the following equation: limit b (P) m−P _(limit) ≤b _(k) +P _(campus,max) ^((p)) ≤P _(limit)where P_(limit) is the power rating of power inverter 106 andP_(campus,max) ^((p)) is the maximum value of P_(campus) at confidencelevel p. This constraint could also be implemented by identifying theprobability that the sum of b_(k) and P_(campus,max) exceeds the powerinverter power rating (e.g., using a probability density function forP_(campus,max)) and limiting that probability to less than or equal to1−p.

In some embodiments, optimization constraints module 414 implements anoptimization constraint to ensure (with a given probability) that theactual SOC of battery 108 remains between zero and one at each time stepduring the applicable frequency response period. This constraint isdifferent from the first optimization constraint which placed bounds onthe expected SOC of battery 108 at the end of each optimization period.The expected SOC of battery 108 can be determined deterministically,whereas the actual SOC of battery 108 is dependent on the campus powerP_(campus) and the actual value of the regulation signal Reg_(signal) ateach time step during the optimization period. In other words, for anyvalue of Reg_(award)>0, there is a chance that battery 108 becomes fullydepleted or fully charged while maintaining the desired power P*_(POI)at POI 110.

Optimization constraints module 414 may implement the constraint on theactual SOC of battery 108 by approximating the battery power P_(bat) (arandom process) as a wide-sense stationary, correlated normallydistributed process. Thus, the SOC of battery 108 can be considered as arandom walk. Determining if the SOC would violate the constraint is anexample of a gambler's ruin problem. For example, consider a random walkdescribed by the following equation:y _(k+1) =y _(k) +x _(k) ,P(x _(k)=1)=p,P(x _(k)=−1)=1−pThe probability P that y_(k) (starting at state z) will reach zero inless than n moves is given by the following equation:

$P = {2{{a^{- 1}\left( {2p} \right)}^{\frac{n - z}{2}}\left\lbrack {2\left( {1 - p} \right)} \right\rbrack}^{\frac{n + z}{2}}{\sum\limits_{v = 1}^{\frac{a}{2}}{{\cos^{n - 1}\left( \frac{\pi\; v}{a} \right)}{\sin\left( \frac{\pi\; v}{a} \right)}{\sin\left( \frac{\pi\;{zv}}{a} \right)}}}}$In some embodiments, each frequency response period includesapproximately n=1800 time steps (e.g., one time step every two secondsfor an hour). Therefore, the central limit theorem applies and it ispossible to convert the autocorrelated random process for P_(bat) andthe limits on the SOC of battery 108 into an uncorrelated random processof 1 or −1 with a limit of zero.

In some embodiments, optimization constraints module 414 converts thebattery power P_(bat) into an uncorrelated normal process driven by theregulation signal Reg_(signal). For example, consider the originalbattery power described by the following equation:x _(k+1) =αx _(k) +e _(k) ,x _(k) ˜N(μ,σ),e _(k) ˜N(μ_(e),σ_(e))where the signal x represents the battery power P_(bat), α is anautocorrelation parameter, and e is a driving signal. In someembodiments, e represents the regulation signal Reg_(signal). If thepower of the signal x is known, then the power of signal e is alsoknown, as shown in the following equations:μ(1−α)=μ_(e)E{x _(k) ²}(1−α)²−2αμμ_(e) =E{e _(k) ²}E{x _(k) ²}(1−α²)−λ²α(1−α)=E{e _(k) ²},Additionally, the impulse response of the difference equation forx_(k+1) is:h _(k)=α^(k) k≥0Using convolution, x_(k) can be expressed as follows:

$x_{k} = {\sum\limits_{i = 1}^{k}{\alpha^{k - i}e_{i - 1}}}$x₃ = α²e₀ + α¹e₁ + e₂x_(q) = α^(q − 1)e₀ + α^(q − 2)e₁ + … + α e_(q − 2) + e_(q − 1)

A random walk driven by signal x_(k) can be defined as follows:

$y_{k} = {{\sum\limits_{j = 1}^{k}x_{j}} = {\sum\limits_{j = 1}^{k}{\sum\limits_{i = 1}^{j}{\alpha^{j - 1}e_{i - 1}}}}}$which for large values of j can be approximated using the infinite sumof a geometric series in terms of the uncorrelated signal e rather thanx:

$y_{k} = {{{\sum\limits_{j = 1}^{k}x_{j}} \approx {\sum\limits_{j = 1}^{k}{\frac{1}{1 - \alpha}e_{j}}}} = {{\sum\limits_{j = 1}^{k}{x_{j}^{\prime}\mspace{14mu} k}} ⪢ 1}}$Thus, the autocorrelated driving signal x_(k) of the random walk can beconverted into an uncorrelated driving signal x′_(k) with mean and powergiven by:

${{E\left\{ x_{k}^{\prime} \right\}} = \mu},{{E\left\{ \left( {x_{k}^{\prime} - \mu} \right)^{2} \right\}} = {\frac{1 + \alpha}{1 - \alpha}\sigma^{2}}},{{E\left\{ {x_{k}^{\prime}}^{2} \right\}} = {{\frac{1 + \alpha}{1 - \alpha}\sigma^{2}} + \mu^{2}}},{\sigma_{x^{\prime}}^{2} = {\frac{1 + \alpha}{1 - \alpha}\sigma^{2}}}$where x′_(k) represents the regulation signal Reg_(signal).Advantageously, this allows optimization constraints module 414 todefine the probability of ruin in terms of the regulation signalReg_(signal).

In some embodiments, optimization constraints module 414 determines aprobability p that the random walk driven by the sequence of −1 and 1will take the value of 1. In order to ensure that the random walk drivenby the sequence of −1 and 1 will behave the same as the random walkdriven by x′_(k), optimization constraints module 414 may select p suchthat the ratio of the mean to the standard deviation is the same forboth driving functions, as shown in the following equations:

$\frac{mean}{stdev} = {\frac{\mu}{\sqrt{\frac{1 + \alpha}{1 - \alpha}\sigma}} = {\overset{\sim}{\mu} = \frac{{2p} - 1}{\sqrt{4{p\left( {1 - p} \right)}}}}}$$p = {\frac{1}{2} \pm {\frac{1}{2}\sqrt{1 - \left( \frac{1}{{\overset{\sim}{\mu}}^{2} + 1} \right)}}}$where {tilde over (μ)} is the ratio of the mean to the standarddeviation of the driving signal (e.g., Reg_(signal)) and μ is the changein state-of-charge over the frequency response period divided by thenumber of time steps within the frequency response period

$\left( {{i.e.},{\mu = \frac{\Delta\;{SOC}}{n}}} \right).$For embodiments in which each frequency response period has a durationof one hour (i.e., 3600 seconds) and the interval between time steps istwo seconds, the number of time steps per frequency response period is1800 (i.e., n=1800). In the equation for p, the plus is used when {tildeover (μ)} is greater than zero, whereas the minus is used when {tildeover (μ)} is less than zero. Optimization constraints module 414 mayalso ensure that both driving functions have the same number of standarddeviations away from zero (or ruin) to ensure that both random walkshave the same behavior, as shown in the following equation:

$z = \frac{{{SOC} \cdot C_{des}}\sqrt{4{p\left( {1 - p} \right)}}}{\sqrt{\frac{1 + \alpha}{1 - \alpha}\sigma}}$

Advantageously, the equations for p and z allow optimization constraintsmodule 414 to define the probability of ruin P (i.e., the probability ofbattery 108 fully depleting or reaching a fully charged state) within Ntime steps (n=1 . . . N) as a function of variables that are known tohigh level controller 312 and/or manipulated by high level controller312. For example, the equation for p defines p as a function of the meanand standard deviation of the regulation signal Reg_(signal), which maybe estimated by signal statistics predictor 318. The equation for zdefines z as a function of the SOC of battery 108 and the parameters ofthe regulation signal Reg_(signal).

Optimization constraints module 414 may use one or more of the previousequations to place constraints on ΔSOC·C_(des) and Reg_(award) given thecurrent SOC of battery 108. For example, optimization constraints module414 may use the mean and standard deviation of the regulation signalReg_(signal) to calculate p. Optimization constraints module 414 maythen use p in combination with the SOC of battery 108 to calculate z.Optimization constraints module 414 may use p and z as inputs to theequation for the probability of ruin P. This allows optimizationconstraints module 414 to define the probability or ruin P as a functionof the SOC of battery 108 and the estimated statistics of the regulationsignal Reg_(signal). Optimization constraints module 414 may imposeconstraints on the SOC of battery 108 to ensure that the probability ofruin P within N time steps does not exceed a threshold value. Theseconstraints may be expressed as limitations on the variablesΔSOC·C_(des) and/or Reg_(award), which are related to midpoint b aspreviously described.

In some embodiments, optimization constraints module 414 uses theequation for the probability of ruin P to define boundaries on thecombination of variables p and z. The boundaries represent thresholdswhen the probability of ruin P in less than N steps is greater than acritical value P_(cr) (e.g., P_(cr)=0.001). For example, optimizationconstraints module 414 may generate boundaries that correspond to athreshold probability of battery 108 fully depleting or reaching a fullycharged state during a frequency response period (e.g., in N=1800steps).

In some embodiments, optimization constraints module 414 constrains theprobability of ruin P to less than the threshold value, which imposeslimits on potential combinations of the variables p and z. Since thevariables p and z are related to the SOC of battery 108 and thestatistics of the regulation signal, the constraints may imposelimitations on ΔSOC·C_(des) and Reg_(award) given the current SOC ofbattery 108. These constraints may also impose limitations on midpoint bsince the variables ΔSOC·C_(des) and Reg_(award) are related to midpointb. For example, optimization constraints module 414 may set constraintson the maximum bid Reg_(award) given a desired change in the SOC forbattery 108. In other embodiments, optimization constraints module 414penalizes the objective function J given the bid Reg_(award) and thechange in SOC.

Still referring to FIG. 4, variable SOC controller 408 is shown toinclude a frequency response (FR) revenue estimator 416. FR revenueestimator 416 may be configured to estimate the frequency responserevenue that will result from a given midpoint b (e.g., a midpointprovided by midpoint optimizer 412). The estimated frequency responserevenue may be used as the term Rev(Reg_(award,k)) in the objectivefunction. Midpoint optimizer 412 may use the estimated frequencyresponse revenue along with other terms in the objective function J todetermine an optimal midpoint b.

In some embodiments, FR revenue estimator 416 uses a revenue model topredict frequency response revenue. An exemplary revenue model which maybe used by FR revenue estimator 416 is shown in the following equation:Rev(Reg_(award))=Reg_(award)(CP_(cap)+MR·CP_(perf))where CP_(cap), MR, and CP_(perf) are the energy market statisticsreceived from energy market predictor 316 and Reg_(award) is a functionof the midpoint b. For example, capability bid calculator 422 maycalculate Reg_(award) using the following equation:Reg_(award) =P _(limit) −|b|where P_(limit) is the power rating of power inverter 106.

As shown above, the equation for frequency response revenue used by FRrevenue estimator 416 does not include a performance score (or assumes aperformance score of 1.0). This results in FR revenue estimator 416estimating a maximum possible frequency response revenue that can beachieved for a given midpoint b (i.e., if the actual frequency responseof controller 112 were to follow the regulation signal exactly).However, it is contemplated that the actual frequency response may beadjusted by low level controller 314 in order to preserve the life ofbattery 108. When the actual frequency response differs from theregulation signal, the equation for frequency response revenue can beadjusted to include a performance score. The resulting value function Jmay then be optimized by low level controller 314 to determine anoptimal frequency response output which considers both frequencyresponse revenue and the costs of battery degradation, as described withreference to FIG. 5.

Still referring to FIG. 4, variable SOC controller 408 is shown toinclude a battery degradation estimator 418. Battery degradationestimator 418 may estimate the cost of battery degradation that willresult from a given midpoint b (e.g., a midpoint provided by midpointoptimizer 412). The estimated battery degradation may be used as theterm λ_(bat) in the objective function J. Midpoint optimizer 412 may usethe estimated battery degradation along with other terms in theobjective function J to determine an optimal midpoint b.

In some embodiments, battery degradation estimator 418 uses a batterylife model to predict a loss in battery capacity that will result from aset of midpoints b, power outputs, and/or other variables that can bemanipulated by controller 112. The battery life model may define theloss in battery capacity C_(loss,add) as a sum of multiple piecewiselinear functions, as shown in the following equation:C _(loss,add) =f ₁(T _(cell))+f ₂(SOC)+f ₃(DOD)+f ₄(PR)+f ₅(ER)−C_(loss,nom)where T_(cell) is the cell temperature, SOC is the state-of-charge, DODis the depth of discharge, PR is the average power ratio

$\left( {{e.g.},{{PR} = {{avg}\left( \frac{P_{avg}}{P_{des}} \right)}}} \right),$and ER is the average effort ratio

$\left( {{e.g.},{{ER} = {{avg}\left( \frac{\Delta\; P_{bat}}{P_{des}} \right)}}} \right.$of battery 108. C_(loss,nom) is the nominal loss in battery capacitythat is expected to occur over time. Therefore, C_(loss,add) representsthe additional loss in battery capacity degradation in excess of thenominal value C_(loss,nom).

In some embodiments, the battery life model is generated by a batterylife model generator 428. Battery life model generator 428 may definethe terms in the battery life model as functions of one or morevariables that have known values (e.g., estimated or measured values)and/or variables that can be manipulated by high level controller 312.For example, battery life model generator 428 may define the terms inthe battery life model as functions of the regulation signal statistics(e.g., the mean and standard deviation of Reg_(signal)), the campuspower signal statistics (e.g., the mean and standard deviation ofP_(campus)), Reg_(award), midpoint b, the SOC of battery 108, and/orother variables that have known or controlled values.

In some embodiments, battery life model generator 428 measures the celltemperature T_(cell) using a temperature sensor configured to measurethe temperature of battery 108. In other embodiments, battery life modelgenerator 428 estimates or predicts the cell temperature T_(cell) basedon a history of measured temperature values. For example, battery lifemodel generator 428 may use a predictive model to estimate the celltemperature T_(cell) as a function of the battery power P_(bat), theambient temperature, and/or other variables that can be measured,estimated, or controlled by high level controller 312.

Battery life model generator 428 may define the variable SOC in thebattery life model as the SOC of battery 108 at the end of the frequencyresponse period. The SOC of battery 108 may be measured or estimatedbased on the control decisions made by controller 112. For example,battery life model generator 428 may use a predictive model to estimateor predict the SOC of battery 108 at the end of the frequency responseperiod as a function of the battery power P_(bat), the midpoint b,and/or other variables that can be measured, estimated, or controlled byhigh level controller 312.

Battery life model generator 428 may define the average power ratio PRas the ratio of the average power output of battery 108 (i.e., P_(avg))to the design power

${P_{des}\left( {{e.g.},{{PR} = \frac{P_{avg}}{P_{des}}}} \right)}.$The average power output of battery 108 can be defined using thefollowing equation:P _(avg) =E{|Reg_(award)Reg_(signal) +b−P _(loss) −P _(campus)|}where the expression (Reg_(award)Reg_(signal)+b−P_(loss)−P_(campus))represents the battery power P_(bat). The expected value of P_(avg) isgiven by:

$P_{avg} = {{\sigma_{bat}\sqrt{\frac{2}{\pi}}{\exp\left( \frac{- \mu_{bat}^{2}}{2\sigma_{bat}^{2}} \right)}} + {{erf}\left( \frac{- \mu_{bat}}{\sqrt{2\sigma_{bat}^{2}}} \right)}}$where μ_(bat) and σ_(bat) ² are the mean and variance of the batterypower P_(bat). The variables μ_(bat) and π_(bat) ² may be defined asfollows:μ_(bat)=Reg_(award) E{Reg_(signal) }+b−E{P _(loss) }−E{P _(campus)}σ_(bat) ²=Reg_(award) ²τ_(FR) ²+σ_(campus) ²where σ_(FR) ² is the variance of Reg_(signal) and the contribution ofthe battery power loss to the variance σ_(bat) ² is neglected.

Battery life model generator 428 may define the average effort ratio ERas the ratio of the average change in battery power ΔP_(avg) to thedesign power

${P_{des}\left( {{i.e.},{{ER} = \frac{\Delta\; P_{bat}}{P_{des}}}} \right)}.$The average change in battery power can be defined using the followingequation:ΔP _(avg) =E{P _(bat,k) −P _(bat,k−1)}ΔP _(avg) =E{|Reg_(award)(Reg_(signal,k)−Reg_(signal,k−1))−(P _(loss,k)−P _(loss,k−1))−(P _(campus,k) −P _(campus,k−1))|}To make this calculation more tractable, the contribution due to thebattery power loss can be neglected. Additionally, the campus powerP_(campus) and the regulation signal Reg_(signal) can be assumed to beuncorrelated, but autocorrelated with first order autocorrelationparameters of α_(campus) and α, respectively. The argument inside theabsolute value in the equation for ΔP_(avg) has a mean of zero and avariance given by:

$\begin{matrix}{\sigma_{diff}^{2} = {E\left\{ \left\lbrack {{{Reg}_{award}\left( {{Reg}_{{signal},k} - {Reg}_{{signal},{k - 1}}} \right)} - \left( {P_{{campus},k} - P_{{campus},{k - 1}}} \right)} \right\rbrack^{2} \right\}}} \\{= {E\left\{ {{{Reg}_{award}^{2}\left( {{Reg}_{{signal},k} - {Reg}_{{signal},{k - 1}}} \right)}^{2} - \left( {P_{{campus},k} - P_{{campus},{k - 1}}} \right)^{2}} \right\}}} \\{= {{2{{Reg}_{award}^{2}\left( {1 - \alpha} \right)}\sigma_{FR}^{2}} + {2\left( {1 - \alpha_{campus}} \right)\sigma_{campus}^{2}}}}\end{matrix}$

Battery life model generator 428 may define the depth of discharge DODas the maximum state-of-charge minus the minimum state-of-charge ofbattery 108 over the frequency response period, as shown in thefollowing equation:DOD=SOC_(max)−SOC_(min)The SOC of battery 108 can be viewed as a constant slope with a zeromean random walk added to it, as previously described. An uncorrelatednormal random walk with a driving signal that has zero mean has anexpected range given by:

${E\left\{ {\max - \min} \right\}} = {2\sigma\sqrt{\frac{2N}{\pi}}}$where E{max−min} represents the depth of discharge DOD and can beadjusted for the autocorrelation of the driving signal as follows:

${E\left\{ {\max - \min} \right\}} = {2\sigma_{bat}\sqrt{\frac{1 + \alpha_{bat}}{1 - \alpha_{bat}}}\sqrt{\frac{2N}{\pi}}}$σ_(bat)² = Reg_(award)²σ_(FR)² + σ_(campus)²$\alpha_{bat} = \frac{{{Reg}_{award}^{2}{\alpha\sigma}_{FR}^{2}} + {\alpha_{campus}\sigma_{campus}^{2}}}{{{Reg}_{award}^{2}\sigma_{FR}^{2}} + \sigma_{campus}^{2}}$

If the SOC of battery 108 is expected to change (i.e., is not zeromean), the following equation may be used to define the depth ofdischarge DOD:

${E\left\{ {\max - \min} \right\}} = \left\{ \begin{matrix}{R_{0} + {{c \cdot \Delta}\;{{SOC} \cdot \exp}\left\{ {{- \alpha}\frac{R_{0} - {\Delta\;{SOC}}}{\sigma_{bat}}} \right\}}} & {{\Delta\;{SOC}} < R_{0}} \\{{\Delta\;{SOC}} + {{c \cdot R_{0} \cdot \exp}\left\{ {{- \alpha}\frac{{\Delta\;{SOC}} - R_{0}}{\sigma_{bat}}} \right\}}} & {{\Delta\;{SOC}} > R_{0}}\end{matrix} \right.$where R₀ is the expected range with zero expected change in thestate-of-charge.Advantageously, battery life model generator 428 may use the previousequations to define each of the variables in the battery life model(e.g., T_(cell), SOC, DOD, PR, and ER) as a function of variables thatare known to controller 112 and/or variables that can be manipulated bycontroller 112 (e.g., battery power setpoint, frequency response controlsignals, ramp rate control signals, etc.).

In some embodiments, battery life model generator 428 defines thebattery life model as a parametric function of the variables T_(cell),SOC, DOD, PR, and ER. For example, battery life model generator 428 mayformulate the battery life model as follows:C _(loss,add)=θ₁ T _(cell)+θ₂SOC+θ₃DOD+θ₄PR+θ₅ER−C _(loss,nom)where θ₁-θ₅ are parameters of the battery life model (e.g., regressioncoefficients) and the variables T_(cell), SOC, DOD, PR, and ER are thesame as previously described.

Battery life model generator 428 may perform a series of experiments todetermine values for the model parameters θ₁-θ₅. For example, batterylife model life generator 428 may apply known battery power signals(e.g., frequency response signals, ramp rate control signals, etc.) topower inverter 106, causing power inverter 106 to charge or dischargebattery 108 according to a known profile. Battery life model generator428 may calculate values for the input variables to the battery lifemodel (e.g., T_(cell), SOC, DOD, PR, and ER) based on the known batterypower signals. Battery life model generator 428 may then measure adecrease in battery capacity that results from the known battery powersignals. From this experiment, battery life model generator 428 canobtain values for all of the input variables to the battery life modeland the resulting output variable C_(loss,add).

Battery life model generator 428 may perform this experiment multipletimes to generate test data. The test data may include multipledifferent sets of values for the input variables T_(cell), SOC, DOD, PR,and ER, along with the resultant value of the output variableC_(loss,add) corresponding to each set of input variables. Battery lifemodel generator 428 may then perform a curve fitting process (e.g., aregression process) to determine values for the model parameters θ₁-θ₅that best fit the test data. Battery life model generator 428 mayprovide the battery life model with the values for the model parametersθ₁-θ₅ to battery degradation estimator 418. Battery degradationestimator 418 may use the battery life model to estimate a loss inbattery capacity that will result from a new set of input variablesbased on an actual or potential set of control outputs (e.g., frequencyresponse signals, ramp rate control signals, battery power controlsignals, etc.). A process for generating and using a battery life modelis described in greater detail with reference to FIG. 9.

Still referring to FIG. 4, variable SOC controller 408 is shown toinclude a revenue loss estimator 420. Revenue loss estimator 420 may beconfigured to estimate an amount of potential revenue that will be lostas a result of the battery capacity loss C_(loss,add). In someembodiments, revenue loss estimator 420 converts battery capacity lossC_(loss,add) into lost revenue using the following equation:R _(loss)=(CP_(cap)+MR·CP_(perf))C _(loss,add) P _(des)where R_(loss) is the lost revenue over the duration of the frequencyresponse period.

Revenue loss estimator 420 may determine a present value of the revenueloss R_(loss) using the following equation:

$\lambda_{bat} = {\left\lbrack \frac{1 - \left( {1 + \frac{i}{n}} \right)^{- n}}{\frac{i}{n}} \right\rbrack R_{loss}}$where n is the total number of frequency response periods (e.g., hours)during which the revenue loss occurs and λ_(bat) is the present value ofthe revenue loss during the ith frequency response period. In someembodiments, the revenue loss occurs over ten years (e.g., n=87,600hours). Revenue loss estimator 420 may provide the present value of therevenue loss λ_(bat) to midpoint optimizer 412 for use in the objectivefunction J.

Midpoint optimizer 412 may use the inputs from optimization constraintsmodule 414, FR revenue estimator 416, battery degradation estimator 418,and revenue loss estimator 420 to define the terms in objective functionJ. Midpoint optimizer 412 may determine values for midpoint b thatoptimize objective function J. In various embodiments, midpointoptimizer 412 may use sequential quadratic programming, dynamicprogramming, or any other optimization technique to optimize theobjective function J.

Still referring to FIG. 4, high level controller 312 is shown to includea capability bid calculator 422. Capability bid calculator 422 may beconfigured to generate a capability bid Reg_(award) based on themidpoint b generated by constant SOC controller 402 and/or variable SOCcontroller 408. In some embodiments, capability bid calculator 422generates a capability bid that is as large as possible for a givenmidpoint, as shown in the following equation:Reg_(award) =P _(limit) −|b|where P_(limit) is the power rating of power inverter 106. Capabilitybid calculator 422 may provide the capability bid to incentive provider114 and to frequency response optimizer 424 for use in generating anoptimal frequency response.Filter Parameters Optimization

Still referring to FIG. 4, high level controller 312 is shown to includea frequency response optimizer 424 and a filter parameters optimizer426. Filter parameters optimizer 426 may be configured to generate a setof filter parameters for low level controller 314. The filter parametersmay be used by low level controller 314 as part of a low-pass filterthat removes high frequency components from the regulation signalReg_(signal). In some embodiments, filter parameters optimizer 426generates a set of filter parameters that transform the regulationsignal Reg_(signal) into an optimal frequency response signal Res_(FR).Frequency response optimizer 424 may perform a second optimizationprocess to determine the optimal frequency response Res_(FR) based onthe values for Reg_(award) and midpoint b. In the second optimization,the values for Reg_(award) and midpoint b may be fixed at the valuespreviously determined during the first optimization.

In some embodiments, frequency response optimizer 424 determines theoptimal frequency response Res_(FR) by optimizing value function J shownin the following equation:

$J = {{\sum\limits_{k = 1}^{h}{{Rev}\left( {Reg}_{{award},k} \right)}} + {\sum\limits_{k = 1}^{h}{c_{k}b_{k}}} + {\min\limits_{period}\left( {P_{{campus},k} + b_{k}} \right)} - {\sum\limits_{k = 1}^{h}\lambda_{{bat},k}}}$where the frequency response revenue Rev(Reg_(award)) is defined asfollows:Rev(Reg_(award))=PS·Reg_(award)(CP_(cap)+MR·CP_(perf))and the frequency response Res_(FR) is substituted for the regulationsignal Reg_(signal) in the battery life model used to calculateλ_(bat,k). The performance score PS may be based on several factors thatindicate how well the optimal frequency response Res_(FR) tracks theregulation signal Reg_(signal).

The frequency response Res_(FR) may affect both Rev(Reg_(award)) and themonetized cost of battery degradation λ_(bat). Closely tracking theregulation signal may result in higher performance scores, therebyincreasing the frequency response revenue. However, closely tracking theregulation signal may also increase the cost of battery degradationλ_(bat). The optimized frequency response Res_(FR) represents an optimaltradeoff between decreased frequency response revenue and increasedbattery life (i.e., the frequency response that maximizes value J).

In some embodiments, the performance score PS is a composite weightingof an accuracy score, a delay score, and a precision score. Frequencyresponse optimizer 424 may calculate the performance score PS using theperformance score model shown in the following equation:PS=−⅓PS_(acc)+⅓PS_(delay)+⅓PS_(prec)where PS_(acc) is the accuracy score, PS_(delay) is the delay score, andPS_(prec) is the precision score. In some embodiments, each term in theprecision score is assigned an equal weighting (e.g., ⅓). In otherembodiments, some terms may be weighted higher than others.

The accuracy score PS_(acc) may be the maximum correlation between theregulation signal Reg_(signal) and the optimal frequency responseRes_(FR). Frequency response optimizer 424 may calculate the accuracyscore PS_(acc) using the following equation:

${PS}_{acc} = {\max\limits_{\delta}\mspace{14mu} r_{{Reg},{{Res}{(\delta)}}}}$where δ is a time delay between zero and δ_(max) (e.g., between zero andfive minutes).

The delay score PS_(delay) may be based on the time delay δ between theregulation signal Reg_(signal) and the optimal frequency responseRes_(FR). Frequency response optimizer 424 may calculate the delay scorePS_(delay) using the following equation:

${PS}_{delay} = {\frac{{\delta\lbrack s\rbrack} - \delta_{\max}}{\delta_{\max}}}$where δ [s] is the time delay of the frequency response Res_(FR)relative to the regulation signal Reg_(signal) and δ_(max) is themaximum allowable delay (e.g., 5 minutes or 300 seconds).

The precision score PS_(prec) may be based on a difference between thefrequency response Res_(FR) and the regulation signal Reg_(signal).Frequency response optimizer 424 may calculate the precision scorePS_(prec) using the following equation:

${PS}_{prec} = {1 - \frac{\sum{{{Res}_{FR} - {Reg}_{signal}}}}{\sum{{Reg}_{signal}}}}$

Frequency response optimizer 424 may use the estimated performance scoreand the estimated battery degradation to define the terms in objectivefunction J. Frequency response optimizer 424 may determine values forfrequency response Res_(FR) that optimize objective function J. Invarious embodiments, frequency response optimizer 424 may use sequentialquadratic programming, dynamic programming, or any other optimizationtechnique.

Filter parameters optimizer 426 may use the optimized frequency responseRes_(FR) to generate a set of filter parameters for low level controller314. In some embodiments, the filter parameters are used by low levelcontroller 314 to translate an incoming regulation signal into afrequency response signal. Low level controller 314 is described ingreater detail with reference to FIG. 5.

In some embodiments, high level controller 312 includes a data fusionmodule 430 (shown in FIG. 14). Data fusion module 430 is configured toaggregate data received from external systems and devices for processingby high level controller 312. For example, data fusion module 430 maystore and aggregate external data such as the campus power signal,utility rates, incentive event history and/or weather forecasts as shownin FIG. 14. Further, data fusion module 430 may store and aggregate datafrom low level controller 314. For example, data fusion module 430 mayreceive data such as battery SOC, battery temperature, battery systemtemperature data, security device status data, battery voltage data,battery current data and/or any other data provided by battery system1304. Data fusion module 430 is described in greater detail withreference to FIG. 14.

Low Level Controller

Referring now to FIG. 5, a block diagram illustrating low levelcontroller 314 in greater detail is shown, according to an exemplaryembodiment. Low level controller 314 may receive the midpoints b and thefilter parameters from high level controller 312. Low level controller314 may also receive the campus power signal from campus 102 and theregulation signal Reg_(signal) and the regulation award Reg_(award) fromincentive provider 114.

Predicting and Filtering the Regulation Signal

Low level controller 314 is shown to include a regulation signalpredictor 502. Regulation signal predictor 502 may use a history of pastand current values for the regulation signal Reg_(signal) to predictfuture values of the regulation signal. In some embodiments, regulationsignal predictor 502 uses a deterministic plus stochastic model trainedfrom historical regulation signal data to predict future values of theregulation signal Reg_(signal). For example, regulation signal predictor502 may use linear regression to predict a deterministic portion of theregulation signal Reg_(signal) and an AR model to predict a stochasticportion of the regulation signal Reg_(signal). In some embodiments,regulation signal predictor 502 predicts the regulation signalReg_(signal) using the techniques described in U.S. patent applicationSer. No. 14/717,593, titled “Building Management System for ForecastingTime Series Values of Building Variables” and filed May 20, 2015.

Low level controller 314 is shown to include a regulation signal filter504. Regulation signal filter 504 may filter the incoming regulationsignal Reg_(signal) and/or the predicted regulation signal using thefilter parameters provided by high level controller 312. In someembodiments, regulation signal filter 504 is a low pass filterconfigured to remove high frequency components from the regulationsignal Reg_(signal). Regulation signal filter 504 may provide thefiltered regulation signal to power setpoint optimizer 506.

Determining Optimal Power Setpoints

Power setpoint optimizer 506 may be configured to determine optimalpower setpoints for power inverter 106 based on the filtered regulationsignal. In some embodiments, power setpoint optimizer 506 uses thefiltered regulation signal as the optimal frequency response. Forexample, low level controller 314 may use the filtered regulation signalto calculate the desired interconnection power P*_(POI) using thefollowing equation:P* _(POI)=Reg_(award)·Reg_(filter) +bwhere Reg_(filter) is the filtered regulation signal. Power setpointoptimizer 506 may subtract the campus power P_(campus) from the desiredinterconnection power P*_(POI) to calculate the optimal power setpointsP_(SP) for power inverter 106, as shown in the following equation:P _(SP) =P* _(POI) −P _(campus)

In other embodiments, low level controller 314 performs an optimizationto determine how closely to track P*_(POI). For example, low levelcontroller 314 is shown to include a frequency response optimizer 508.Frequency response optimizer 508 may determine an optimal frequencyresponse Res_(FR) by optimizing value function J shown in the followingequation:

$J = {{\sum\limits_{k = 1}^{h}{{Rev}\left( {Reg}_{{award},k} \right)}} + {\sum\limits_{k = 1}^{h}{c_{k}b_{k}}} + {\min\limits_{period}\left( {P_{{campus},k} + b_{k}} \right)} - {\sum\limits_{k = 1}^{h}\lambda_{{bat},k}}}$where the frequency response Res_(FR) affects both Rev(Reg_(award)) andthe monetized cost of battery degradation λ_(bat). The frequencyresponse Res_(FR) may affect both Rev(Reg_(award)) and the monetizedcost of battery degradation λ_(bat). The optimized frequency responseRes_(FR) represents an optimal tradeoff between decreased frequencyresponse revenue and increased battery life (i.e., the frequencyresponse that maximizes value J). The values of Rev(Reg_(award)) andλ_(bat,k) may be calculated by FR revenue estimator 510, performancescore calculator 512, battery degradation estimator 514, and revenueloss estimator 516.Estimating Frequency Response Revenue

Still referring to FIG. 5, low level controller 314 is shown to includea FR revenue estimator 510. FR revenue estimator 510 may estimate afrequency response revenue that will result from the frequency responseRes_(FR). In some embodiments, FR revenue estimator 510 estimates thefrequency response revenue using the following equation:Rev(Reg_(award))=PS·Reg_(award)(CP_(cap)+MR·CP_(perf))where Reg_(award), CP_(cap), MR, and CP_(perf) are provided as knowninputs and PS is the performance score.

Low level controller 314 is shown to include a performance scorecalculator 512. Performance score calculator 512 may calculate theperformance score PS used in the revenue function. The performance scorePS may be based on several factors that indicate how well the optimalfrequency response Res_(FR) tracks the regulation signal Reg_(signal).In some embodiments, the performance score PS is a composite weightingof an accuracy score, a delay score, and a precision score. Performancescore calculator 512 may calculate the performance score PS using theperformance score model shown in the following equation:PS=⅓PS_(acc)+⅓PS_(delay)+⅓PS_(prec)where PS_(acc) is the accuracy score, PS_(delay) is the delay score, andPS_(prec) is the precision score. In some embodiments, each term in theprecision score is assigned an equal weighting (e.g., ⅓). In otherembodiments, some terms may be weighted higher than others. Each of theterms in the performance score model may be calculated as previouslydescribed with reference to FIG. 4.Estimating Battery Degradation

Still referring to FIG. 5, low level controller 314 is shown to includea battery degradation estimator 514. Battery degradation estimator 514may be the same or similar to battery degradation estimator 418, withthe exception that battery degradation estimator 514 predicts thebattery degradation that will result from the frequency responseRes_(FR) rather than the original regulation signal Reg_(signal). Theestimated battery degradation may be used as the term λ_(batt) in theobjective function J. Frequency response optimizer 508 may use theestimated battery degradation along with other terms in the objectivefunction J to determine an optimal frequency response Res_(FR).

In some embodiments, battery degradation estimator 514 uses a batterylife model to predict a loss in battery capacity that will result fromthe frequency response Res_(FR). The battery life model may define theloss in battery capacity C_(loss,add) as a sum of multiple piecewiselinear functions, as shown in the following equation:C _(loss,add) =f ₁(T _(cell))+f ₂(SOC)+f ₃(DOD)+f ₄(PR)+f ₅(ER)−C_(loss,nom)where T_(cell) is the cell temperature, SOC is the state-of-charge, DODis the depth of discharge, PR is the average power ratio

$\left( {{e.g.},{{PR} = {{avg}\left( \frac{P_{avg}}{P_{des}} \right)}}} \right),$and ER is the average effort ratio

$\left( {{e.g.},{{ER} = {{avg}\left( \frac{\Delta\; P_{bat}}{P_{des}} \right)}}} \right.$of battery 108. C_(loss,nom) is the nominal loss in battery capacitythat is expected to occur over time. Therefore, C_(loss,add) representsthe additional loss in battery capacity degradation in excess of thenominal value C_(loss,nom). The terms in the battery life model may becalculated as described with reference to FIG. 4, with the exceptionthat the frequency response Res_(FR) is used in place of the regulationsignal Reg_(signal). In some embodiments, battery degradation estimator514 uses the battery life model generated by battery life modelgenerator 428 to estimate the loss in battery capacity.

Still referring to FIG. 5, low level controller 314 is shown to includea revenue loss estimator 516. Revenue loss estimator 516 may be the sameor similar to revenue loss estimator 420, as described with reference toFIG. 4. For example, revenue loss estimator 516 may be configured toestimate an amount of potential revenue that will be lost as a result ofthe battery capacity loss C_(loss,add). In some embodiments, revenueloss estimator 516 converts battery capacity loss C_(loss,add) into lostrevenue using the following equation:R _(loss)=(CP_(cap)+MR·CP_(perf))C _(loss,add) P _(des)where R_(loss) is the lost revenue over the duration of the frequencyresponse period.

Revenue loss estimator 420 may determine a present value of the revenueloss R_(loss) using the following equation:

$\lambda_{bat} = {\left\lbrack \frac{1 - \left( {1 + \frac{i}{n}} \right)^{- n}}{\frac{i}{n}} \right\rbrack R_{loss}}$where n is the total number of frequency response periods (e.g., hours)during which the revenue loss occurs and Δ_(bat) is the present value ofthe revenue loss during the ith frequency response period. In someembodiments, the revenue loss occurs over ten years (e.g., n=87,600hours). Revenue loss estimator 420 may provide the present value of therevenue loss λ_(bat) to frequency response optimizer 508 for use in theobjective function J.

Frequency response optimizer 508 may use the estimated performance scoreand the estimated battery degradation to define the terms in objectivefunction J. Frequency response optimizer 508 may determine values forfrequency response Res_(FR) that optimize objective function J. Invarious embodiments, frequency response optimizer 508 may use sequentialquadratic programming, dynamic programming, or any other optimizationtechnique.

Frequency Response Optimization Processes

Referring now to FIG. 6, a flowchart of a process 600 for determiningoptimal battery power setpoints in a frequency response optimizationsystem is shown, according to an exemplary embodiment. Process 600 maybe performed by one or more components of frequency responseoptimization system 100, as described with reference to FIGS. 1-5.

Process 600 is shown to include predicting regulation signal statistics,campus power use, and energy market statistics for one or more frequencyresponse periods (step 602). The regulation signal statistics mayinclude a mean and a standard deviation of the regulation signalReg_(signal). Predicting the campus power use may include predicting theelectric power consumption of the campus (e.g., P_(campus)) over theduration of the frequency response periods. In some embodiments,predicting campus power use includes predicting statistics of the campuspower use (e.g., mean, standard deviation, etc.). Predicting energymarket statistics may include predicting energy prices such aselectricity rates and demand charges. Predicting energy marketstatistics may also include predicting frequency response statisticssuch as clearing capability price CP_(cap), clearing performance priceCP_(perf), and mileage ratio MR.

Process 600 is shown to include determining frequency response midpointsthat maintain a battery at the same state-of-charge at the beginning andend of each frequency response period (step 604). Step 604 may beperformed by constant state-of-charge controller 402. In someembodiments, step 604 includes determining expected values for theregulation signal Reg_(signal), campus power use P_(campus), regulationaward Reg_(award), battery power loss P_(loss), and maximum batterypower P_(max). The expected value of the regulation award Reg_(award)may be assumed to be the difference between the midpoint b and the powerinverter power limit P_(limit) (e.g., Reg_(award)=P_(limit)−|b|). Themidpoint b which maintains the battery at the same state-of-charge atthe beginning and end of each frequency response period can bedetermined by solving the following equation:

${{\frac{1}{4P_{\max}}\left\lbrack {{E\left\{ P_{campus}^{2} \right\}} + {{Reg}_{award}^{2}E\left\{ {Reg}_{siginal}^{2} \right\}} - {2{Reg}_{award}E\left\{ {Reg}_{siginal} \right\} E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} + {\quad{{{\left\lbrack {{{Reg}_{award}E\left\{ {Reg}_{siginal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack\Delta\; t} + {{\frac{b}{2P_{\max}}\left\lbrack {{{Reg}_{award}E\left\{ {Reg}_{siginal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} + {b\;\Delta\; t} + {\frac{b^{2}}{4P_{\max}}\Delta\; t}} = 0}}$which can be solved for midpoint b.

Process 600 is shown to include generating an objective function thatestimates value as a function of frequency response revenue, energycosts, and monetized costs of losses in battery capacity (step 606). Anexemplary value function which may be generated in step 606 is shown inthe following equation:

$J = {{\sum\limits_{k = 1}^{h}{{Rev}\left( {Reg}_{{award},k} \right)}} + {\sum\limits_{k = 1}^{h}{c_{k}b_{k}}} + {\min\limits_{period}\left( {P_{{campus},k} + b_{k}} \right)} - {\sum\limits_{k = 1}^{h}\lambda_{{bat},k}}}$where Rev(Reg_(award,k)) is the frequency response revenue, c_(k)b_(k)is the cost per unit of electricity, the min( ) term is the demandcharge, and λ_(bat,k) is the monetized cost of battery degradation.

The frequency response revenue may be determined using the followingequation:Rev(Reg_(award))=PS·Reg_(award)(CP_(cap)+MR·CP_(perf))where Reg_(award), CP_(cap), MR, and CP_(perf) are provided as knowninputs and PS is the performance score. In some embodiments, theperformance score PS is a composite weighting of an accuracy score, adelay score, and a precision score. The performance score PS may becalculated using the performance score model shown in the followingequation:PS=⅓PS_(acc)+⅓PS_(delay)+⅓PS_(prec)where PS_(acc) is the accuracy score, PS_(delay) is the delay score, andPS_(prec) is the precision score. In some embodiments, each term in theprecision score is assigned an equal weighting (e.g., ⅓). In otherembodiments, some terms may be weighted higher than others. Each of theterms in the performance score model may be calculated as previouslydescribed with reference to FIG. 4.

The cost of battery degradation may be determined using a battery lifemodel. An exemplary battery life model which may be used to determinebattery degradation is shown in the following equation:C _(loss,add) =f ₁(T _(cell))+f ₂(SOC)+f ₃(DOD)+f ₄(PR)+f ₅(ER)−C_(loss,nom)where T_(cell) is the cell temperature, SOC is the state-of-charge, DODis the depth of discharge, PR is the average power ratio

$\left( {{e.g.},{{PR} = {{avg}\left( \frac{P_{avg}}{P_{des}} \right)}}} \right),$and ER is the average effort ratio

$\left( {{e.g.},{{ER} = {{avg}\left( \frac{\Delta\; P_{bat}}{P_{des}} \right)}}} \right.$of battery 108. The terms in the battery life model may be calculated asdescribed with reference to FIGS. 4-5, using the frequency responseRes_(FR) as the optimization variable.

The battery capacity loss C_(loss,add) can be converted into lostrevenue using the following equation:R _(loss)=(CP_(cap)+MR·CP_(perf))C _(loss,add) P _(des)where R_(loss) is the lost revenue over the duration of the frequencyresponse period. The present value of the revenue loss R_(loss) can becalculated as follows:

$\lambda_{bat} = {\left\lbrack \frac{1 - \left( {1 + \frac{i}{n}} \right)^{- n}}{\frac{i}{n}} \right\rbrack R_{loss}}$where n is the total number of frequency response periods (e.g., hours)during which the revenue loss occurs and λ_(bat) is the present value ofthe revenue loss during the ith frequency response period. In someembodiments, the revenue loss occurs over ten years (e.g., n=87,600hours).

Still referring to FIG. 6, process 600 is shown to include performing anoptimization to determine battery power setpoints that optimize theobjective function based on the midpoints (step 608). Step 608 may beperformed by low level controller 314, as described with reference toFIG. 5. Step 608 may include using the objective function to determinethe optimal frequency response Res_(FR). The optimal frequency responsemay be the frequency response which maximizes J. The optimal frequencyresponse Res_(FR) can be used to calculate the desired interconnectionpower P*_(POI) using the following equation:P* _(POI)=Reg_(award)·Res_(FR) +b

Step 608 may include subtracting the campus power P_(campus) from thedesired interconnection power P*_(POI) to calculate the optimal batterypower setpoints P_(SP), as shown in the following equation:P _(SP) =P* _(POI) −P _(campus)

Process 600 is shown to include using the battery power setpoints tocontrol an amount of power charged or discharged from the battery (step610). Step 610 may include providing the battery power setpoints to abattery power inverter (e.g., power inverter 106). The power invertermay use the battery power setpoints to an amount of power drawn from thebattery or stored in the battery during each of a plurality of timesteps. Power drawn from the battery may be used to power the campus orprovided to an energy grid.

Referring now to FIG. 7, a flowchart of a process 700 for determiningoptimal battery power setpoints in a frequency response optimizationsystem is shown, according to an exemplary embodiment. Process 700 maybe performed by one or more components of frequency responseoptimization system 100, as described with reference to FIGS. 1-5.

Process 700 is shown to include predicting regulation signal statistics,campus power use, and energy market statistics for one or more frequencyresponse periods (step 702). The regulation signal statistics mayinclude a mean and a standard deviation of the regulation signalReg_(signal). Predicting the campus power use may include predicting theelectric power consumption of the campus (e.g., P_(campus)) over theduration of the frequency response periods. In some embodiments,predicting campus power use includes predicting statistics of the campuspower use (e.g., mean, standard deviation, etc.). Predicting energymarket statistics may include predicting energy prices such aselectricity rates and demand charges. Predicting energy marketstatistics may also include predicting frequency response statisticssuch as clearing capability price CP_(cap), clearing performance priceCP_(perf), and mileage ratio MR.

Process 700 is shown to include generating an objective function thatestimates value as a function of frequency response revenue, energycosts, and monetized costs of losses in battery capacity (step 704). Anexemplary value function which may be generated in step 704 is shown inthe following equation:

$J = {{\sum\limits_{k = 1}^{h}{{Rev}\left( {Reg}_{{award},k} \right)}} + {\sum\limits_{k = 1}^{h}{c_{k}b_{k}}} + {\min\limits_{period}\left( {P_{{campus},k} + b_{k}} \right)} - {\sum\limits_{k = 1}^{h}\lambda_{{bat},k}}}$where Rev(Reg_(award,k)) is the frequency response revenue, c_(k)b_(k)is the cost per unit of electricity, the min( ) term is the demandcharge, and λ_(bat,k) is the monetized cost of battery degradation.

The frequency response revenue may be determined using the followingequation:Rev(Reg_(award))=Reg_(award)(CP_(cap)+MR·CP_(perf))where Reg_(award), CP_(cap), MR, and CP_(perf) are provided as knowninputs and PS is the performance score.

The cost of battery degradation may be determined using a battery lifemodel. An exemplary battery life model which may be used to determinebattery degradation is shown in the following equation:C _(loss,add) =f ₁(T _(cell))+f ₂(SOC)+f ₃(DOD)+f ₄(PR)+f ₅(ER)−C_(loss,nom)where T_(cell) is the cell temperature, SOC is the state-of-charge, DODis the depth of discharge, PR is the average power ratio

$\left( {{e.g.},{{PR} = {{avg}\left( \frac{P_{avg}}{P_{des}} \right)}}} \right),$and ER is the average effort ratio

$\left( {{e.g.},{{ER} = {{avg}\left( \frac{\Delta\; P_{bat}}{P_{des}} \right)}}} \right.$of battery 108. The terms in the battery life model may be calculated asdescribed with reference to FIGS. 4-5, using the midpoint b as theoptimization variable.

The battery capacity loss C_(loss,add) can be converted into lostrevenue using the following equation:R _(loss)=(CP_(cap)+MR·CP_(perf))C _(loss,add) P _(des)where R_(loss) is the lost revenue over the duration of the frequencyresponse period. The present value of the revenue loss R_(loss) can becalculated as follows:

$\lambda_{bat} = {\left\lbrack \frac{1 - \left( {1 + \frac{i}{n}} \right)^{- n}}{\frac{i}{n}} \right\rbrack R_{loss}}$where n is the total number of frequency response periods (e.g., hours)during which the revenue loss occurs and λ_(bat) is the present value ofthe revenue loss during the ith frequency response period. In someembodiments, the revenue loss occurs over ten years (e.g., n=87,600hours).

Process 700 is shown to include performing a high level optimization todetermine frequency response midpoints that optimize the objectivefunction based on the predicted regulation signal statistics (step 706).Step 706 may be performed by variable state-of-charge controller 408.Step 706 may include determining a set of midpoints b that maximize theobjective function J. The optimization may be subject to optimizationconstraints. For example, step 706 may include implementing optimizationconstraints that keep the state-of-charge of the battery betweenthreshold values (e.g., between 0 and 1) and/or constraints that preventsum of the midpoint b and the campus power P_(campus) from exceeding thepower inverter rating P_(limit), as described with respect tooptimization constraints module 414.

In some embodiments, step 706 includes determining expected values forthe regulation signal Reg_(signal), campus power use P_(campus),regulation award Reg_(award), battery power loss P_(loss), and maximumbattery power P_(max). The expected value of the regulation awardReg_(award) may be assumed to be the difference between the midpoint band the power inverter power limit P_(limit) (e.g.,Reg_(award)=P_(limit)−|b|). The midpoint b can be expressed in terms ofthe change in the state-of-charge of the battery by solving thefollowing equation:

${{\frac{1}{4P_{\max}}\left\lbrack {{E\left\{ P_{campus}^{2} \right\}} + {Reg}_{award}^{2} + {E\left\{ {Reg}_{siginal}^{2} \right\}} - {2{Reg}_{award}E\left\{ {Reg}_{siginal} \right\} E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} + {\quad{{{\left\lbrack {{{Reg}_{award}E\left\{ {Reg}_{siginal} \right\}} + {E\left\{ P_{campus} \right\}}} \right\rbrack\Delta\; t} + {\Delta\;{{SOC} \cdot C_{des}}} + {{\frac{b}{2P_{\max}}\left\lbrack {{{Reg}_{award}E\left\{ {Reg}_{signal} \right\}} - {E\left\{ P_{campus} \right\}}} \right\rbrack}\Delta\; t} + {b\;\Delta\; t} + {\frac{b^{2}}{4P_{\max}}\Delta\; t}} = 0}}$This relationship between midpoint b and the change in the SOC may beused to translate constraints on the SOC of the battery into constraintson midpoint b. The optimization problem can then be solved to determineoptimal values for midpoint b.

Still referring to FIG. 7, process 700 is shown to include performing alow level optimization to determine battery power setpoints thatoptimize the objective function based on the midpoints (step 708). Step708 may be performed by low level controller 314, as described withreference to FIG. 5. Step 708 may include using the objective functionto determine the optimal frequency response Res_(FR). The optimalfrequency response may be the frequency response which maximizes J. Theoptimal frequency response Res_(FR) can be used to calculate the desiredinterconnection power P*_(POI) using the following equation:P* _(POI)=Reg_(award)·Res_(FR) +b

Step 708 may include subtracting the campus power P_(campus) from thedesired interconnection power P*_(POI) to calculate the optimal batterypower setpoints P_(SP), as shown in the following equation:P _(SP) =P* _(POI) −P _(campus)

Process 700 is shown to include using the battery power setpoints tocontrol an amount of power charged or discharged from the battery (step710). Step 710 may include providing the battery power setpoints to abattery power inverter (e.g., power inverter 106). The power invertermay use the battery power setpoints to an amount of power drawn from thebattery or stored in the battery during each of a plurality of timesteps. Power drawn from the battery may be used to power the campus orprovided to an energy grid.

Photovoltaic Energy System with Frequency Regulation and Ramp RateControl

Referring now to FIGS. 8A-8B, a photovoltaic energy system 800 that usesbattery storage to simultaneously perform both ramp rate control andfrequency regulation is shown, according to an exemplary embodiment. Aspreviously described, frequency regulation is the process of maintainingthe stability of the grid frequency (e.g., 60 Hz in the United States).The grid frequency may remain balanced at 60 Hz as long as there is abalance between the demand from the energy grid and the supply to theenergy grid. An increase in demand yields a decrease in grid frequency,whereas an increase in supply yields an increase in grid frequency.During a fluctuation of the grid frequency, system 800 may offset thefluctuation by either drawing more energy from the energy grid (e.g., ifthe grid frequency is too high) or by providing energy to the energygrid (e.g., if the grid frequency is too low).

Ramp rate control is the process of offsetting PV ramp rates (i.e.,increases or decreases in PV power output) that fall outside ofcompliance limits determined by the electric power authority overseeingthe energy grid. Ramp rate control typically requires the use of anenergy source that allows for offsetting ramp rates by either supplyingadditional power to the grid or consuming more power from the grid.Advantageously, system 800 may use battery storage in combination withphotovoltaic power to perform frequency regulation while simultaneouslycomplying with ramp rate requirements and maintaining thestate-of-charge of the battery storage within a predetermined desirablerange. System 800 may use the battery life model described withreference to FIGS. 1-7 to make decisions regarding battery usage.

System 800 is shown to include a photovoltaic (PV) field 802, a PV fieldpower inverter 804, a battery 806, a battery power inverter 808, a pointof interconnection (POI) 810, and an energy grid 812. In someembodiments, system 800 also includes a controller 814 (shown in FIG.8A) and/or a building 818 (shown in FIG. 8B). In brief overview, PVfield 802 may convert solar energy into a DC power output P_(PV) andprovide the DC power output P_(PV) to PV field power inverter 804. PVfield power inverter 804 may convert the DC power output P_(PV) into anAC power output u_(PV) and provide the AC power output u_(PV) to POI810. PV field power inverter 804 can be operated by controller 814 tocontrol the power output of PV field 802. Similarly, battery powerinverter 808 can be operated by controller 814 to control the powerinput and/or power output of battery 806. The power outputs of PV fieldpower inverter 804 and battery power inverter 808 combine at POI 810 toform the power provided to energy grid 812. In some embodiments,building 818 is also connected to POI 810. Building 818 can consume aportion of the combined power at POI 810 to satisfy the energyrequirements of building 818.

PV field power inverter 804 can include any of a variety of circuitcomponents (e.g., resistors, capacitors, indictors, transformers,transistors, switches, diodes, etc.) configured to perform the functionsdescribed herein. In some embodiments DC power from PV field 802 isconnected to a transformer of PV field power inverter 804 through acenter tap of a primary winding. A switch can be rapidly switched backand forth to allow current to flow back to PV field 802 following twoalternate paths through one end of the primary winding and then theother. The alternation of the direction of current in the primarywinding of the transformer can produce alternating current (AC) in asecondary circuit.

In some embodiments, PV field power inverter 804 uses anelectromechanical switching device to convert DC power from PV field 802into AC power. The electromechanical switching device can include twostationary contacts and a spring supported moving contact. The springcan hold the movable contact against one of the stationary contacts,whereas an electromagnet can pull the movable contact to the oppositestationary contact. Electric current in the electromagnet can beinterrupted by the action of the switch so that the switch continuallyswitches rapidly back and forth. In some embodiments, PV field powerinverter 804 uses transistors, thyristors (SCRs), and/or various othertypes of semiconductor switches to convert DC power from PV field 802into AC power. SCRs provide large power handling capability in asemiconductor device and can readily be controlled over a variablefiring range.

In some embodiments, PV field power inverter 804 produces a squarevoltage waveform (e.g., when not coupled to an output transformer). Inother embodiments, PV field power inverter 804 produces a sinusoidalwaveform that matches the sinusoidal frequency and voltage of energygrid 812. For example, PV field power inverter 804 can use Fourieranalysis to produce periodic waveforms as the sum of an infinite seriesof sine waves. The sine wave that has the same frequency as the originalwaveform is called the fundamental component. The other sine waves,called harmonics, that are included in the series have frequencies thatare integral multiples of the fundamental frequency.

In some embodiments, PV field power inverter 804 uses inductors and/orcapacitors to filter the output voltage waveform. If PV field powerinverter 804 includes a transformer, filtering can be applied to theprimary or the secondary side of the transformer or to both sides.Low-pass filters can be applied to allow the fundamental component ofthe waveform to pass to the output while limiting the passage of theharmonic components. If PV field power inverter 804 is designed toprovide power at a fixed frequency, a resonant filter can be used. If PVfield power inverter 804 is an adjustable frequency inverter, the filtercan be tuned to a frequency that is above the maximum fundamentalfrequency. In some embodiments, PV field power inverter 804 includesfeedback rectifiers or antiparallel diodes connected acrosssemiconductor switches to provide a path for a peak inductive loadcurrent when the switch is turned off. The antiparallel diodes can besimilar to freewheeling diodes commonly used in AC/DC convertercircuits.

Battery power inverter 808 may be configured to draw a DC power P_(bat)from battery 806, convert the DC power P_(bat) into an AC power u_(bat),and provide the AC power u_(bat) to POI 810. Battery power inverter 808may also be configured to draw the AC power u_(bat) from POI 810,convert the AC power u_(bat) into a DC battery power P_(bat), and storethe DC battery power P_(bat) in battery 806. As such, battery powerinverter 808 can function as both a power inverter and a rectifier toconvert between DC and AC in either direction. The DC battery powerP_(bat) may be positive if battery 806 is providing power to batterypower inverter 808 (i.e., if battery 806 is discharging) or negative ifbattery 806 is receiving power from battery power inverter 808 (i.e., ifbattery 806 is charging). Similarly, the AC battery power u_(bat) may bepositive if battery power inverter 808 is providing power to POI 810 ornegative if battery power inverter 808 is receiving power from POI 810.

Like PV field power inverter 804, battery power inverter 808 can includeany of a variety of circuit components (e.g., resistors, capacitors,indictors, transformers, transistors, switches, diodes, etc.) configuredto perform the functions described herein. Battery power inverter 808can include many of the same components as PV field power inverter 804and can operate using similar principles. For example, battery powerinverter 808 can use electromechanical switching devices, transistors,thyristors (SCRs), and/or various other types of semiconductor switchesto convert between AC and DC power. Battery power inverter 808 canoperate the circuit components to adjust the amount of power stored inbattery 806 and/or discharged from battery 806 (i.e., power throughput)based on a power control signal or power setpoint from controller 814.

The AC battery power u_(bat) is shown to include an amount of power usedfor frequency regulation (i.e., u_(FR)) and an amount of power used forramp rate control (i.e., u_(RR)) which together form the AC batterypower (i.e., u_(bat)=u_(FR)+u_(RR)). The DC battery power P_(bat) isshown to include both u_(FR) and u_(RR) as well as an additional termP_(loss) representing power losses in battery 806 and/or battery powerinverter 808 (i.e., P_(bat)=u_(FR)+u_(RR)+P_(loss)). The PV field poweru_(PV) and the battery power u_(bat) combine at POI 810 to form P_(POI)(i.e., P_(POI)=u_(PV)+u_(bat)), which represents the amount of powerprovided to energy grid 812. P_(POI), may be positive if POI 810 isproviding power to energy grid 812 or negative if POI 810 is receivingpower from energy grid 812.

Still referring to FIGS. 8A-8B, system 800 is shown to include acontroller 814. Controller 814 may be configured to generate a PV powersetpoint u_(PV) for PV field power inverter 804 and a battery powersetpoint u_(bat) for battery power inverter 808. Throughout thisdisclosure, the variable u_(PV) is used to refer to both the PV powersetpoint generated by controller 814 and the AC power output of PV fieldpower inverter 804 since both quantities have the same value. Similarly,the variable u_(bat) is used to refer to both the battery power setpointgenerated by controller 814 and the AC power output/input of batterypower inverter 808 since both quantities have the same value.

PV field power inverter 804 uses the PV power setpoint u_(PV) to controlan amount of the PV field power P_(PV) to provide to POI 810. Themagnitude of u_(PV) may be the same as the magnitude of P_(pV) or lessthan the magnitude of P_(PV). For example, u_(PV) may be the same asP_(PV) if controller 814 determines that PV field power inverter 804 isto provide all of the photovoltaic power P_(PV) to POI 810. However,u_(PV) may be less than P_(PV) if controller 814 determines that PVfield power inverter 804 is to provide less than all of the photovoltaicpower P_(PV) to POI 810. For example, controller 814 may determine thatit is desirable for PV field power inverter 804 to provide less than allof the photovoltaic power P_(PV) to POI 810 to prevent the ramp ratefrom being exceeded and/or to prevent the power at POI 810 fromexceeding a power limit.

Battery power inverter 808 uses the battery power setpoint u_(bat) tocontrol an amount of power charged or discharged by battery 806. Thebattery power setpoint u_(bat) may be positive if controller 814determines that battery power inverter 808 is to draw power from battery806 or negative if controller 814 determines that battery power inverter808 is to store power in battery 806. The magnitude of u_(bat) controlsthe rate at which energy is charged or discharged by battery 806.

Controller 814 may generate u_(PV) and u_(bat) based on a variety ofdifferent variables including, for example, a power signal from PV field802 (e.g., current and previous values for P_(PV)), the currentstate-of-charge (SOC) of battery 806, a maximum battery power limit, amaximum power limit at POI 810, the ramp rate limit, the grid frequencyof energy grid 812, and/or other variables that can be used bycontroller 814 to perform ramp rate control and/or frequency regulation.Advantageously, controller 814 generates values for u_(PV) and u_(bat)that maintain the ramp rate of the PV power within the ramp ratecompliance limit while participating in the regulation of grid frequencyand maintaining the SOC of battery 806 within a predetermined desirablerange.

In some embodiments, controller 814 uses a battery life model todetermine optimal battery power setpoints u_(bat). For example,controller 814 may include a battery degradation estimator (e.g.,battery degradation estimator 418 or 514) configured to estimate anamount of battery degradation that will result from the battery powersetpoints u_(bat). Controller 814 may use the battery life modelproduced by battery life model generator 428 to estimate a decrease inbattery capacity that will result from a planned or actual set ofcontrol outputs (e.g., a frequency regulation control signal, a ramprate control signal, a battery power signal, etc.). Controller 814 mayinclude a revenue loss estimator (e.g., revenue loss estimator 420 or516) configured to estimate an amount of potential revenue that will belost as a result of the battery degradation. Controller 814 may adjustthe frequency regulation setpoints u_(FR), the ramp rate controlsetpoints u_(RR), and/or the battery power setpoints u_(bat) in order tooptimize an objective function that considers frequency responserevenue, ramp rate compliance penalties, and the monetized cost ofbattery degradation, as described with reference to FIGS. 4-5.

Reactive Ramp Rate Control

Controller 814 may be configured to control a ramp rate of the poweroutput 816 provided to energy grid 812. Ramp rate may be defined as thetime rate of change of power output 816. Power output 816 may varydepending on the magnitude of the DC output provided by PV field 802.For example, if a cloud passes over PV field 802, power output 816 mayrapidly and temporarily drop while PV field 802 is within the cloud'sshadow. Controller 814 may be configured to calculate the ramp rate bysampling power output 816 and determining a change in power output 816over time. For example, controller 814 may calculate the ramp rate asthe derivative or slope of power output 816 as a function of time, asshown in the following equations:

${{Ramp}\mspace{14mu}{Rate}} = {{\frac{dP}{dt}\mspace{14mu}{or}\mspace{14mu}{Ramp}\mspace{14mu}{Rate}} = \frac{\Delta\; P}{\Delta\; t}}$where P represents power output 816 and t represents time.

In some embodiments, controller 814 controls the ramp rate to complywith regulatory requirements or contractual requirements imposed byenergy grid 812. For example, system 800 may be required to maintain theramp rate within a predetermined range in order to deliver power toenergy grid 812. In some embodiments, system 800 is required to maintainthe absolute value of the ramp rate at less than a threshold value(e.g., less than 10% of the rated power capacity per minute). In otherwords, system 800 may be required to prevent power output 816 fromincreasing or decreasing too rapidly. If this requirement is not met,system 800 may be deemed to be in non-compliance and its capacity may bede-rated, which directly impacts the revenue generation potential ofsystem 800.

Controller 814 may use battery 806 to perform ramp rate control. Forexample, controller 814 may use energy from battery 806 to smooth asudden drop in power output 816 so that the absolute value of the ramprate is less than a threshold value. As previously mentioned, a suddendrop in power output 816 may occur when a solar intensity disturbanceoccurs, such as a passing cloud blocking the sunlight to PV field 802.Controller 814 may use the energy from battery 806 to make up thedifference between the power provided by PV field 802 (which hassuddenly dropped) and the minimum required power output 816 to maintainthe required ramp rate. The energy from battery 806 allows controller814 to gradually decrease power output 816 so that the absolute value ofthe ramp rate does not exceed the threshold value.

Once the cloud has passed, the power output from PV field 802 maysuddenly increase as the solar intensity returns to its previous value.Controller 814 may perform ramp rate control by gradually ramping uppower output 816. Ramping up power output 816 may not require energyfrom battery 806. For example, power inverter 804 may use only a portionof the energy generated by PV field 802 (which has suddenly increased)to generate power output 816 (i.e., limiting the power output) so thatthe ramp rate of power output 816 does not exceed the threshold value.The remainder of the energy generated by PV field 802 (i.e., the excessenergy) may be stored in battery 806 and/or dissipated. Limiting theenergy generated by PV field 802 may include diverting or dissipating aportion of the energy generated by PV field 802 (e.g., using variableresistors or other circuit elements) so that only a portion of theenergy generated by PV field 802 is provided to energy grid 812. Thisallows power inverter 804 to ramp up power output 816 gradually withoutexceeding the ramp rate. The excess energy may be stored in battery 806,used to power other components of system 800, or dissipated.

Referring now to FIG. 10, a graph 1000 illustrating a reactive ramp ratecontrol technique which can be used by system 800 is shown, according toan exemplary embodiment. Graph 1000 plots the power output P provided toenergy grid 812 as a function of time t. The solid line 1002 illustratespower output P without any ramp rate control, whereas the broken line1004 illustrates power output P with ramp rate control.

Between times t₀ and t₁, power output P is at a high value P_(high). Attime t₁, a cloud begins to cast its shadow on PV field 802, causing thepower output of PV field 802 to suddenly decrease, until PV field 802 iscompletely in shadow at time t₂. Without any ramp rate control, thesudden drop in power output from PV field 802 causes the power output Pto rapidly drop to a low value P_(low) at time t₂. However, with ramprate control, system 800 uses energy from battery 806 to graduallydecrease power output P to P_(low) at time t₃. Triangular region 1006represents the energy from battery 806 used to gradually decrease poweroutput P.

Between times t₂ and t₄, PV field 802 is completely in shadow. At timet₄, the shadow cast by the cloud begins to move off PV field 802,causing the power output of PV field 802 to suddenly increase, until PVfield 802 is entirely in sunlight at time t₅. Without any ramp ratecontrol, the sudden increase in power output from PV field 802 causesthe power output P to rapidly increase to the high value P_(high) attime t₅. However, with ramp rate control, power inverter 804 limits theenergy from PV field 802 to gradually increase power output P toP_(high) at time t₆. Triangular region 1008 represents the energygenerated by PV field 802 in excess of the ramp rate limit. The excessenergy may be stored in battery 806 and/or dissipated in order togradually increase power output P at a rate no greater than the maximumallowable ramp rate.

Notably, both triangular regions 1006 and 1008 begin after a change inthe power output of PV field 802 occurs. As such, both the decreasingramp rate control and the increasing ramp rate control provided bysystem 800 are reactionary processes triggered by a detected change inthe power output. In some embodiments, a feedback control technique isused to perform ramp rate control in system 800. For example, controller814 may monitor power output 816 and determine the absolute value of thetime rate of change of power output 816 (e.g., dP/dt or ΔP/Δt).Controller 814 may initiate ramp rate control when the absolute value ofthe time rate of change of power output 816 exceeds a threshold value.

Preemptive Ramp Rate Control

In some embodiments, controller 814 is configured to predict when solarintensity disturbances will occur and may cause power inverter 804 toramp down the power output 816 provided to energy grid 812 preemptively.Instead of reacting to solar intensity disturbances after they occur,controller 814 can actively predict solar intensity disturbances andpreemptively ramp down power output 816 before the disturbances affectPV field 802. Advantageously, this allows system controller 814 toperform both ramp down control and ramp up control by using only aportion of the energy provided by PV field 802 to generate power output816 while the power output of PV field 802 is still high, rather thanrelying on energy from a battery. The remainder of the energy generatedby PV field 802 (i.e., the excess energy) may be stored in battery 806and/or dissipated.

In some embodiments, controller 814 predicts solar intensitydisturbances using input from one or more cloud detectors. The clouddetectors may include an array of solar intensity sensors. The solarintensity sensors may be positioned outside PV field 802 or within PVfield 802. Each solar intensity sensor may have a known location. Insome embodiments, the locations of the solar intensity sensors are basedon the geometry and orientation of PV field 802. For example, if PVfield 802 is rectangular, more sensors may be placed along its long sidethan along its short side. A cloud formation moving perpendicular to thelong side may cover more area of PV field 802 per unit time than a cloudformation moving perpendicular to the short side. Therefore, it may bedesirable to include more sensors along the long side to more preciselydetect cloud movement perpendicular to the long side. As anotherexample, more sensors may be placed along the west side of PV field 802than along the east side of PV field 802 since cloud movement from westto east is more common than cloud movement from east to west. Theplacement of sensors may be selected to detect approaching cloudformations without requiring unnecessary or redundant sensors.

The solar intensity sensors may be configured to measure solar intensityat various locations outside PV field 802. When the solar intensitymeasured by a particular solar intensity sensor drops below a thresholdvalue, controller 814 may determine that a cloud is currently casting ashadow on the solar intensity sensor. Controller 814 may use input frommultiple solar intensity sensors to determine various attributes ofclouds approaching PV field 802 and/or the shadows produced by suchclouds. For example, if a shadow is cast upon two or more of the solarintensity sensors sequentially, controller 814 may use the knownpositions of the solar intensity sensors and the time interval betweeneach solar intensity sensor detecting the shadow to determine how fastthe cloud/shadow is moving. If two or more of the solar intensitysensors are within the shadow simultaneously, controller 814 may use theknown positions of the solar intensity sensors to determine a position,size, and/or shape of the cloud/shadow.

Although the cloud detectors are described primarily as solar intensitysensors, it is contemplated that the cloud detectors may include anytype of device configured to detect the presence of clouds or shadowscast by clouds. For example, the cloud detectors may include one or morecameras that capture visual images of cloud movement. The cameras may beupward-oriented cameras located below the clouds (e.g., attached to astructure on the Earth) or downward-oriented cameras located above theclouds (e.g., satellite cameras). Images from the cameras may be used todetermine cloud size, position, velocity, and/or other cloud attributes.In some embodiments, the cloud detectors include radar or othermeteorological devices configured to detect the presence of clouds,cloud density, cloud velocity, and/or other cloud attributes. In someembodiments, controller 814 receives data from a weather service thatindicates various cloud attributes.

Advantageously, controller 814 may use the attributes of theclouds/shadows to determine when a solar intensity disturbance (e.g., ashadow) is approaching PV field 802. For example, controller 814 may usethe attributes of the clouds/shadows to determine whether any of theclouds are expected to cast a shadow upon PV field 802. If a cloud isexpected to cast a shadow upon PV field 802, controller 814 may use thesize, position, and/or velocity of the cloud/shadow to determine aportion of PV field 802 that will be affected. The affected portion ofPV field 802 may include some or all of PV field 802. Controller 814 mayuse the attributes of the clouds/shadows to quantify a magnitude of theexpected solar intensity disturbance (e.g., an expected decrease inpower output from PV field 802) and to determine a time at which thedisturbance is expected to occur (e.g., a start time, an end time, aduration, etc.).

In some embodiments, controller 814 predicts a magnitude of thedisturbance for each of a plurality of time steps. Controller 814 mayuse the predicted magnitudes of the disturbance at each of the timesteps to generate a predicted disturbance profile. The predicteddisturbance profile may indicate how fast power output 816 is expectedto change as a result of the disturbance. Controller 814 may compare theexpected rate of change to a ramp rate threshold to determine whetherramp rate control is required. For example, if power output 816 ispredicted to decrease at a rate in excess of the maximum compliant ramprate, controller 814 may preemptively implement ramp rate control togradually decrease power output 816.

In some embodiments, controller 814 identifies the minimum expectedvalue of power output 816 and determines when the predicted power outputis expected to reach the minimum value. Controller 814 may subtract theminimum expected power output 816 from the current power output 816 todetermine an amount by which power output 816 is expected to decrease.

Controller 814 may apply the maximum allowable ramp rate to the amountby which power output 816 is expected to decrease to determine a minimumtime required to ramp down power output 816 in order to comply with themaximum allowable ramp rate. For example, controller 814 may divide theamount by which power output 816 is expected to decrease (e.g., measuredin units of power) by the maximum allowable ramp rate (e.g., measured inunits of power per unit time) to identify the minimum time required toramp down power output 816. Controller 814 may subtract the minimumrequired time from the time at which the predicted power output isexpected to reach the minimum value to determine when to startpreemptively ramping down power output 816.

Advantageously, controller 814 may preemptively act upon predicteddisturbances by causing power inverter 804 to ramp down power output 816before the disturbances affect PV field 802. This allows power inverter804 to ramp down power output 816 by using only a portion of the energygenerated by PV field 802 to generate power output 816 (i.e., performingthe ramp down while the power output is still high), rather thanrequiring additional energy from a battery (i.e., performing the rampdown after the power output has decreased). The remainder of the energygenerated by PV field 802 (i.e., the excess energy) may be stored inbattery 806 and/or dissipated.

Referring now to FIG. 11, a graph 1100 illustrating a preemptive ramprate control technique which can be used by controller 814 is shown,according to an exemplary embodiment. Graph 1100 plots the power outputP provided to energy grid 812 as a function of time t. The solid line1102 illustrates power output P without any ramp rate control, whereasthe broken line 1104 illustrates power output P with preemptive ramprate control.

Between times t₀ and t₂, power output P is at a high value P_(high). Attime t₂, a cloud begins to cast its shadow on PV field 802, causing thepower output of PV field 802 to suddenly decrease, until PV field 802 iscompletely in shadow at time t₃. Without any ramp rate control, thesudden drop in power output from PV field 802 causes the power output Pto rapidly drop from P_(high) to a low value P_(low) between times t₂and t₃. However, with preemptive ramp rate control, controller 814preemptively causes power inverter 804 to begin ramping down poweroutput P at time t₁, prior to the cloud casting a shadow on PV field802. The preemptive ramp down occurs between times t₁ and t₃, resultingin a ramp rate that is relatively more gradual. Triangular region 1106represents the energy generated by PV field 802 in excess of the ramprate limit. The excess energy may be limited by power inverter 804and/or stored in battery 806 to gradually decrease power output P at arate no greater than the ramp rate limit.

Between times t₃ and t₄, PV field 802 is completely in shadow. At timet₄, the shadow cast by the cloud begins to move off PV field 802,causing the power output of PV field 802 to suddenly increase, until PVfield 802 is entirely in sunlight at time t₅. Without any ramp ratecontrol, the sudden increase in power output from PV field 802 causesthe power output P to rapidly increase to the high value P_(high) attime t₅. However, with ramp rate control, power inverter 804 uses only aportion of the energy from PV field 802 to gradually increase poweroutput P to P_(high) at time t6. Triangular region 1108 represents theenergy generated by PV field 802 in excess of the ramp rate limit. Theexcess energy may be limited by power inverter 804 and/or stored inbattery 806 to gradually increase power output P at a rate no greaterthan the ramp rate limit.

Notably, a significant portion of triangular region 1106 occurs betweentimes t₁ and t₂, before the disturbance affects PV field 802. As such,the decreasing ramp rate control provided by system 800 is a preemptiveprocess triggered by detecting an approaching cloud, prior to the cloudcasting a shadow upon PV field 802. In some embodiments, controller 814uses a predictive control technique (e.g., feedforward control, modelpredictive control, etc.) to perform ramp down control in system 800.For example, controller 814 may actively monitor the positions, sizes,velocities, and/or other attributes of clouds/shadows that couldpotentially cause a solar intensity disturbance affecting PV field 802.When an approaching cloud is detected at time t₁, controller 814 maypreemptively cause power inverter 804 to begin ramping down power output816. This allows power inverter 804 to ramp down power output 816 bylimiting the energy generated by PV field 802 while the power output isstill high, rather than requiring additional energy from a battery toperform the ramp down once the power output has dropped.

Process for Generating and Using a Battery Life Model

Referring now to FIG. 9, a flowchart of a process 900 for generating andusing a battery life model is shown, according to an exemplaryembodiment. Process 900 may be performed by one or more components offrequency response optimization system 100 (e.g., controller 112) and/orphotovoltaic energy system 800 (e.g., controller 814), as described withreference to FIGS. 1-8B.

Process 900 is shown to include generating a battery life model thatincludes a plurality of variables that depend on battery power setpoints(step 902). The battery life model may define a loss in battery capacityC_(loss,add) as a sum of multiple piecewise linear functions, as shownin the following equation:C _(loss,add) =f ₁(T _(cell))+f ₂(SOC)+f ₃(DOD)+f ₄(PR)+f ₅(ER)−C_(loss,nom)where T_(cell) is the cell temperature, SOC is the state-of-charge, DODis the depth of discharge, PR is the average power ratio

$\left( {{e.g.},{{PR} = {{avg}\left( \frac{P_{avg}}{P_{des}} \right)}}} \right),$and ER is the average effort ratio

$\left( {{e.g.},{{ER} = {{avg}\left( \frac{\Delta\; P_{bat}}{P_{des}} \right)}}} \right.$of the battery.

Process 900 is shown to include identifying a relationship between eachof the variables in the battery life model and the battery powersetpoints (step 904). One or more of the variables in the battery lifemodel (e.g., T_(cell), SOC, DOD, PR, and ER) may a function of variablesthat are known and/or manipulated by a controller responsible forgenerating the battery power setpoints. Advantageously, this allows thecontroller to calculate values for each of the variables in the batterylife model based on known or controlled values. For example, the celltemperature T_(cell) may be measured using a temperature sensor. Thebattery state-of-charge SOC may represent the state-of-charge of thebattery at the end of the frequency response period and can be measuredor estimated based on the control decisions (e.g., charge rates,discharge rates, etc.) made by the controller.

The average power ratio PR may be defined as the ratio of the averagepower output of the battery (i.e., P_(avg)) to the design power

${P_{des}\left( {{e.g.},{{PR} = \frac{P_{avg}}{P_{des}}}} \right)}.$The average power output of the battery can be defined using thefollowing equation:P _(avg) =E{|Reg_(award)Reg_(signal) +b−P _(loss) −P _(campus)|}where the expression (Reg_(award)Reg_(signal)+b−P_(loss)−P_(campus))represents the battery power P_(bat). The expected value of P_(avg) isgiven by:

$P_{avg} = {{\sigma_{bat}\sqrt{\frac{2}{\pi}}{\exp\left( \frac{- \mu_{bat}^{2}}{2\sigma_{bat}^{2}} \right)}} + {{erf}\left( \frac{- \mu_{bat}}{\sqrt{2\sigma_{bat}^{2}}} \right)}}$where μ_(bat) and σ_(bat) ² are the mean and variance of the batterypower P_(bat). The variables μ_(bat) and σ_(bat) ² may be defined asfollows:μ_(bat)=Reg_(award) E{Reg_(signal) }+b−E{P _(loss) }−E{P _(campus)}σ_(bat) ²−Reg_(award) ²σ_(FR) ²+σ_(campus) ²where σ_(FR) ² is the variance of Reg_(signal) and the contribution ofthe battery power loss to the variance σ_(bat) ² is neglected.

The average effort ratio ER may be defined as the ratio of the averagechange in battery power ΔP_(avg) to the design power

${P_{des}\left( {{i.e.},{{ER} = \frac{\Delta\; P_{avg}}{P_{des}}}} \right)}.$The average change in battery power can be defined using the followingequation:ΔP _(avg) =E{P _(bat,k) −P _(bat,k−1)}ΔP _(avg) =E{|Reg_(award)(Reg_(signal,k)−Reg_(signal,k−1))−(P _(loss,k)−P _(loss,k−1))−(P _(campus,k) −P _(campus,k−1))|}To make this calculation more tractable, the contribution due to thebattery power loss can be neglected. Additionally, the campus powerP_(campus) and the regulation signal Reg_(signal) can be assumed to beuncorrelated, but autocorrelated with first order autocorrelationparameters of α_(campus) and α, respectively. The argument inside theabsolute value in the equation for ΔP_(avg) has a mean of zero and avariance given by:

$\begin{matrix}{\sigma_{diff}^{2} = {E\left\{ \left\lbrack {{{Reg}_{award}\left( {{Reg}_{{signal},k} - {Reg}_{{signal},{k - 1}}} \right)} - \left( {P_{{campus},k} - P_{{campus},{k - 1}}} \right)} \right\rbrack^{2} \right\}}} \\{= {E\left\{ {{{Reg}_{award}^{2}\left( {{Reg}_{{signal},k} - {Reg}_{{signal},{k - 1}}} \right)}^{2} - \left( {P_{{campus},k} - P_{{campus},{k - 1}}} \right)^{2}} \right\}}} \\{= {{2\;{{Reg}_{award}^{2}\left( {1 - \alpha} \right)}\sigma_{FR}^{2}} + {2\left( {1 - \alpha_{campus}} \right)\sigma_{campus}^{2}}}}\end{matrix}$

The depth of discharge DOD may be defined as the maximum state-of-chargeminus the minimum state-of-charge of the battery over the frequencyresponse period, as shown in the following equation:DOD=SOC_(max)−SOC_(min)The SOC of the battery can be viewed as a constant slope with a zeromean random walk added to it, as previously described. An uncorrelatednormal random walk with a driving signal that has zero mean has anexpected range given by:

${E\left\{ {\max - \min} \right\}} = {2\;\sigma\sqrt{\frac{2\; N}{\pi}}}$where E{max−min} represents the depth of discharge DOD and can beadjusted for the autocorrelation of the driving signal as follows:

${E\left\{ {\max - \min} \right\}} = {2\;\sigma_{bat}\sqrt{\frac{1 + \alpha_{bat}}{1 - \alpha_{bat}}}\sqrt{\frac{2\; N}{\pi}}}$σ_(bat)² = Reg_(award)²σ_(FR)² + σ_(campus)²$\alpha_{bat} = \frac{{{Reg}_{award}^{2}\alpha\;\sigma_{FR}^{2}} + {\alpha_{campus}\sigma_{campus}^{2}}}{{{Reg}_{award}^{2}\sigma_{FR}^{2}} + \sigma_{campus}^{2}}$

If the SOC of the battery is expected to change (i.e., is not zeromean), the following equation may be used to define the depth ofdischarge DOD:

${E\left\{ {\max - \min} \right\}} = \left\{ \begin{matrix}{R_{0} + {{c \cdot \Delta}\;{{SOC} \cdot \exp}\left\{ {{- \alpha}\frac{R_{0} - {\Delta\;{SOC}}}{\sigma_{bat}}} \right\}}} & {{\Delta\;{SOC}} < R_{0}} \\{{\Delta\;{SOC}} + {{c \cdot R_{0} \cdot \exp}\left\{ {{- \alpha}\frac{{\Delta\;{SOC}} - R_{0}}{\sigma_{bat}}} \right\}}} & {{\Delta\;{SOC}} > R_{0}}\end{matrix} \right.$where R₀ is the expected range with zero expected change in thestate-of-charge. Advantageously, the previous equations can be used todefine each of the variables in the battery life model (e.g., T_(cell),SOC, DOD, PR, and ER) as a function of variables that are known to thecontroller and/or variables that can be manipulated by the controller(e.g., battery power setpoint, frequency response control signals, ramprate control signals, etc.).

In some embodiments, step 904 includes defining the battery life modelas a parametric function of the variables T_(cell), SOC, DOD, PR, andER. For example, the battery life model may be formulated as follows:C _(loss,add)=θ₁ T _(cell)+θ₂SOC+θ₃DOD+θ₄PR+θ₅ER−C _(loss,nom)where θ₁−θ₆ are parameters of the battery life model (e.g., regressioncoefficients) and the variables T_(cell), SOC, DOD, PR, and ER are thesame as previously described.

Step 904 may include performing a series of experiments to determinevalues for the model parameters θ₁-θ₅. For example, step 904 may includeapplying known battery power signals (e.g., frequency response signals,ramp rate control signals, etc.) to the battery power inverter, causingthe power inverter to charge or discharge the battery according to aknown profile. Step 904 may include calculating values for the inputvariables to the battery life model (e.g., T_(cell), SOC, DOD, PR, andER) based on the known battery power signals and measuring a decrease inbattery capacity that results from the known battery power signals. Fromthis experiment, values can be obtained for all of the input variablesto the battery life model and the resulting output variableC_(loss,add). This experiment can be performed multiple times togenerate test data. The test data may include multiple different sets ofvalues for the input variables T_(cell), SOC, DOD, PR, and ER, alongwith the resultant value of the output variable C_(loss,add)corresponding to each set of input variables. Step 904 may includeperforming curve fitting process (e.g., a regression process) todetermine values for the model parameters θ₁-θ₅ that best fit the testdata.

Still referring to FIG. 9, process 900 is shown to include using thebattery life model to estimate an amount of battery degradation thatwill result from the battery power setpoints (step 906). Step 906 mayinclude calculating values for each of the variables in the battery lifemodel based on the battery power setpoints and using the calculatedvariables as inputs to the battery life model. The battery life modelmay provide the estimated amount of battery degradation C_(loss,add) asa function of the input variables.

In some embodiments, step 906 includes estimating an amount of potentialrevenue that will be lost as a result of the battery capacity lossC_(loss,add). In some embodiments, the battery capacity lossC_(loss,add) can be converted into lost revenue using the followingequation:R _(loss)=(CP_(cap)+MR·CP_(perf))C _(loss,add) P _(des)where R_(loss) is the lost revenue over the duration of the frequencyresponse period. A present value of the revenue loss R_(loss) can becalculated using the following equation:

$\lambda_{bat} = {\left\lbrack \frac{1 - \left( {1 + \frac{i}{n}} \right)^{- n}}{\frac{i}{n}} \right\rbrack R_{loss}}$where n is the total number of frequency response periods (e.g., hours)during which the revenue loss occurs and λ_(bat) is the present value ofthe revenue loss during the ith frequency response period. In someembodiments, the revenue loss occurs over ten years (e.g., n=87,600hours).

Process 900 is shown to include adjusting the battery power setpointsbased on the estimated amount of battery degradation (step 908). Step908 may include selecting battery power setpoints that result in anoptimal tradeoff between frequency response revenue and batterydegradation. In some embodiments, the optimal battery power setpointsmaximize an objective function that includes frequency response revenue,costs of electricity, and costs of battery degradation. For example, anobjective function J can be written as:

$J = {{\sum\limits_{k = 1}^{h}{{Rev}\left( {Reg}_{{award},k} \right)}} + {\sum\limits_{k = 1}^{h}{c_{k}b_{k}}} + {\min\limits_{period}\left( {P_{{campus},k} + b_{k}} \right)} - {\sum\limits_{k = 1}^{h}\lambda_{{batt},k}}}$where Rev(Reg_(award,k)) is the frequency response revenue at time stepk, c_(k)b_(k) is the cost of electricity purchased at time step k,

$\min\limits_{period}\left( {P_{{campus},k} + b_{k}} \right)$is the demand charge based on the maximum rate of electricityconsumption during the applicable demand charge period, and λ_(batt,k)is the monetized cost battery degradation at time step k. Step 908 mayinclude performing an optimization process to determine the batterypower setpoints that optimize objective function J.

Process 900 is shown to include using the adjusted battery powersetpoints to control an amount of electric power stored or dischargedfrom the battery to an energy grid (step 910). Step 910 may includeproviding the battery power setpoints to a battery power inverter. Thepower inverter may use the battery power setpoints to control an amountof power stored in the battery or discharged from the battery. Powerstored in the battery may be removed from the energy grid for purposesof ramp rate control and/or frequency regulation. Similarly, powerdischarged from the battery may be provided to the energy grid forpurposes of ramp rate control and/or frequency regulation.

Frequency Regulation and Ramp Rate Controller

Referring now to FIG. 12, a block diagram illustrating controller 814 ingreater detail is shown, according to an exemplary embodiment.Controller 814 is shown to include a communications interface 1202 and aprocessing circuit 1204. Communications interface 1202 may include wiredor wireless interfaces (e.g., jacks, antennas, transmitters, receivers,transceivers, wire terminals, etc.) for conducting data communicationswith various systems, devices, or networks. For example, communicationsinterface 802 may include an Ethernet card and port for sending andreceiving data via an Ethernet-based communications network and/or aWiFi transceiver for communicating via a wireless communicationsnetwork. Communications interface 1202 may be configured to communicatevia local area networks or wide area networks (e.g., the Internet, abuilding WAN, etc.) and may use a variety of communications protocols(e.g., BACnet, IP, LON, etc.).

Communications interface 1202 may be a network interface configured tofacilitate electronic data communications between controller 814 andvarious external systems or devices (e.g., PV field 802, energy grid812, PV field power inverter 804, battery power inverter 808, etc.). Forexample, controller 814 may receive a PV power signal from PV field 802indicating the current value of the PV power P_(PV) generated by PVfield 802. Controller 814 may use the PV power signal to predict one ormore future values for the PV power P_(PV) and generate a ramp ratesetpoint u_(RR). Controller 814 may receive a grid frequency signal fromenergy grid 812 indicating the current value of the grid frequency.Controller 814 may use the grid frequency to generate a frequencyregulation setpoint u_(FR). Controller 814 may use the ramp ratesetpoint u_(RR) and the frequency regulation setpoint u_(FR) to generatea battery power setpoint u_(bat) and may provide the battery powersetpoint u_(bat) to battery power inverter 808. Controller 814 may usethe battery power setpoint u_(bat) to generate a PV power setpointu_(PV) and may provide the PV power setpoint u_(PV) to PV field powerinverter 804.

Still referring to FIG. 12, processing circuit 1204 is shown to includea processor 1206 and memory 1208. Processor 1206 may be a generalpurpose or specific purpose processor, an application specificintegrated circuit (ASIC), one or more field programmable gate arrays(FPGAs), a group of processing components, or other suitable processingcomponents.

Processor 1206 may be configured to execute computer code orinstructions stored in memory 1208 or received from other computerreadable media (e.g., CDROM, network storage, a remote server, etc.).

Memory 1208 may include one or more devices (e.g., memory units, memorydevices, storage devices, etc.) for storing data and/or computer codefor completing and/or facilitating the various processes described inthe present disclosure. Memory 1208 may include random access memory(RAM), read-only memory (ROM), hard drive storage, temporary storage,non-volatile memory, flash memory, optical memory, or any other suitablememory for storing software objects and/or computer instructions. Memory1208 may include database components, object code components, scriptcomponents, or any other type of information structure for supportingthe various activities and information structures described in thepresent disclosure. Memory 1208 may be communicably connected toprocessor 1206 via processing circuit 1204 and may include computer codefor executing (e.g., by processor 1206) one or more processes describedherein.

Predicting PV Power Output

Still referring to FIG. 12, controller 814 is shown to include a PVpower predictor 1212. PV power predictor 1212 may receive the PV powersignal from PV field 802 and use the PV power signal to make a shortterm prediction of the photovoltaic power output P_(PV). In someembodiments, PV power predictor 1212 predicts the value of P_(PV) forthe next time step (i.e., a one step ahead prediction). For example, ateach time step k, PV power predictor 1212 may predict the value of thePV power output P_(PV) for the next time step k+1 (i.e., {circumflexover (P)}_(PV)(k+1)). Advantageously, predicting the next value for thePV power output P_(PV) allows controller 814 to predict the ramp rateand perform an appropriate control action to prevent the ramp rate fromexceeding the ramp rate compliance limit.

In some embodiments, PV power predictor 1212 performs a time seriesanalysis to predict {circumflex over (P)}_(PV)(k+1). A time series maybe defined by an ordered sequence of values of a variable at equallyspaced intervals. PV power predictor 1212 may model changes betweenvalues of P_(PV) over time using an autoregressive moving average (ARMA)model or an autoregressive integrated moving average (ARIMA) model. PVpower predictor 1212 may use the model to predict the next value of thePV power output P_(PV) and correct the prediction using a Kalman filtereach time a new measurement is acquired. The time series analysistechnique is described in greater detail in the following paragraphs.

In some embodiments, PV power predictor 1212 uses a technique in theBox-Jenkins family of techniques to perform the time series analysis.These techniques are statistical tools that use past data (e.g., lags)to predict or correct new data, and other techniques to find theparameters or coefficients of the time series. A general representationof a time series from the Box-Jenkins approach is:

${X_{k} - {\sum\limits_{r = 1}^{p}{\varphi_{r}X_{k - r}}}} = {\sum\limits_{s = 0}^{q}{\theta_{s}\epsilon_{k - s}}}$which is known as an ARMA process. In this representation, theparameters p and q define the order and number of lags of the timeseries, φ is an autoregressive parameter, and θ is a moving averageparameter. This representation is desirable for a stationary processwhich has a mean, variance, and autocorrelation structure that does notchange over time. However, if the original process {Y_(k)} representingthe time series values of P_(PV) is not stationary, X_(k) can representthe first difference (or higher order difference) of the process{Y_(k)−Y_(k−1)}. If the difference is stationary, PV power predictor1212 may model the process as an ARIMA process.

PV power predictor 1212 may be configured to determine whether to use anARMA model or an ARIMA model to model the time series of the PV poweroutput P_(PV). Determining whether to use an ARMA model or an ARIMAmodel may include identifying whether the process is stationary. In someembodiments, the power output P_(PV) is not stationary. However, thefirst difference Y_(k)−Y_(k1) may be stationary. Accordingly, PV powerpredictor 1212 may select an ARIMA model to represent the time series ofP_(PV).

PV power predictor 1212 may find values for the parameters p and q thatdefine the order and the number of lags of the time series. In someembodiments, PV power predictor 1212 finds values for p and q bychecking the partial autocorrelation function (PACF) and selecting anumber where the PACF approaches zero (e.g., p=q). For some time seriesdata, PV power predictor 1212 may determine that a 4^(th) or 5^(th)order model is appropriate. However, it is contemplated that PV powerpredictor 1212 may select a different model order to represent differenttime series processes.

PV power predictor 1212 may find values for the autoregressive parameterφ_(1 . . . p) and the moving average parameter θ_(1 . . . q). In someembodiments, PV power predictor 1212 uses an optimization algorithm tofind values for φ_(1 . . . p) and θ_(1 . . . p) given the time seriesdata {Y_(k)}. For example, PV power predictor 1212 may generate adiscrete-time ARIMA model of the form:

${{A(z)}{y(k)}} = {\left\lbrack \frac{C(z)}{1 - z^{- 1}} \right\rbrack{e(t)}}$where A(z) and C(z) are defined as follows:A(z)=1+φ₁ z ⁻¹+φ₂ z ⁻²+φ₃ z ⁻³+φ₄ z ⁻⁴C(z)=1+θ₁ z ⁻¹+θ₂ z ⁻²+θ₃ z ⁻³+θ₄ z ⁻⁴where the values for φ_(1 . . . p) and θ_(1 . . . p) are determined byfitting the model to the time series values of P_(PV).

In some embodiments, PV power predictor 1212 uses the ARIMA model as anelement of a Kalman filter. The Kalman filter may be used by PV powerpredictor 1212 to correct the estimated state and provide tighterpredictions based on actual values of the PV power output P_(PV). Inorder to use the ARIMA model with the Kalman filter, PV power predictor1212 may generate a discrete-time state-space representation of theARIMA model of the form:x(k+1)=Ax(k)+Ke(k)y(k)=Cx(k)+e(k)where y(k) represents the values of the PV power output P_(PV) and e(k)is a disturbance considered to be normal with zero mean and a variancederived from the fitted model. It is contemplated that the state-spacemodel can be represented in a variety of different forms. For example,the ARIMA model can be rewritten as a difference equation and used togenerate a different state-space model using state-space modelingtechniques. In various embodiments, PV power predictor 1212 may use anyof a variety of different forms of the state-space model.

The discrete Kalman filter consists of an iterative process that takes astate-space model and forwards it in time until there are available datato correct the predicted state and obtain a better estimate. Thecorrection may be based on the evolution of the mean and covariance ofan assumed white noise system. For example, PV power predictor 1212 mayuse a state-space model of the following form:x(k+1)=Ax(k)+Bu(k)+w(k) w(k)˜N(0,Q)y(k)=Cx(k)+Du(k)+v(k) v(k)˜N(0,R)where N( ) represents a normal distribution, v(k) is the measurementerror having zero mean and variance R, and w(k) is the process errorhaving zero mean and variance Q. The values of R and Q are designchoices. The variable x(k) is a state of the process and the variabley(k) represents the PV power output P_(PV)(k). This representation isreferred to as a stochastic state-space representation.

PV power predictor 1212 may use the Kalman filter to perform aniterative process to predict {circumflex over (P)}_(PV)(k+1) based onthe current and previous values of P_(PV) (e.g., P_(PV)(k), P_(PV)(k−1),etc.). The iterative process may include a prediction step and an updatestep. The prediction step moves the state estimate forward in time usingthe following equations:{circumflex over (x)} ⁻(k+1)=A*{circumflex over (x)}(k)P ⁻(k+1)=A*P(k)*A ^(T) +Qwhere {circumflex over (x)}(k) is the mean of the process or estimatedstate at time step k and P(k) is the covariance of the process at timestep k. The super index “−” indicates that the estimated state{circumflex over (x)}⁻(k+1) is based on the information known prior totime step k+1 (i.e., information up to time step k). In other words, themeasurements at time step k+1 have not yet been incorporated to generatethe state estimate {circumflex over (x)}⁻(k+1). This is known as an apriori state estimate.

PV power predictor 1212 may predict the PV power output {circumflex over(P)}_(PV) (k+1) by determining the value of the predicted measurementŷ⁻(k+1). As previously described, the measurement y(k) and the statex(k) are related by the following equation:y(k)=Cx(k)+e(k)which allows PV power predictor 1212 to predict the measurement ŷ⁻(k+1)as a function of the predicted state {circumflex over (x)}⁻(k+1). PVpower predictor 1212 may use the measurement estimate ŷ⁻(k+1) as thevalue for the predicted PV power output {circumflex over (P)}_(PV)(k+1)(i.e., {circumflex over (P)}_(PV)(k+1)=ŷ⁻(k+1)).

The update step uses the following equations to correct the a prioristate estimate {circumflex over (x)}⁻(k+1) based on the actual(measured) value of y(k+1):K=P ⁻(k+1)*C ^(T)*[R+C*P ⁻(k+1)*C ^(T)]⁻¹{circumflex over (x)}(k+1)={circumflex over (x)}⁻(k+1)+K[y(k+1)−C*{circumflex over (x)} ⁻(k+1)]P(k+1)=P ⁻(k+1)−K*[R+C*P ⁻(k+1)*C ^(T)]*K ^(T)where y(k+1) corresponds to the actual measured value of P_(PV)(k+1).The variable {circumflex over (x)}(k+1) represents the a posterioriestimate of the state x at time k+1 given the information known up totime step k+1. The update step allows PV power predictor 1212 to preparethe Kalman filter for the next iteration of the prediction step.

Although PV power predictor 1212 is primarily described as using a timeseries analysis to predict {circumflex over (P)}_(PV) (k+1), it iscontemplated that PV power predictor 1212 may use any of a variety oftechniques to predict the next value of the PV power output P_(PV). Forexample, PV power predictor 1212 may use a deterministic plus stochasticmodel trained from historical PV power output values (e.g., linearregression for the deterministic portion and an AR model for thestochastic portion). This technique is described in greater detail inU.S. patent application Ser. No. 14/717,593, titled “Building ManagementSystem for Forecasting Time Series Values of Building Variables” andfiled May 20, 2015, the entirety of which is incorporated by referenceherein.

In other embodiments, PV power predictor 1212 uses input from clouddetectors (e.g., cameras, light intensity sensors, radar, etc.) topredict when an approaching cloud will cast a shadow upon PV field 802.When an approaching cloud is detected, PV power predictor 1212 mayestimate an amount by which the solar intensity will decrease as aresult of the shadow and/or increase once the shadow has passed PV field802. PV power predictor 1212 may use the predicted change in solarintensity to predict a corresponding change in the PV power outputP_(PV). This technique is described in greater detail in U.S.Provisional Patent Application No. 62/239,131 titled “Systems andMethods for Controlling Ramp Rate in a Photovoltaic Energy System” andfiled Oct. 8, 2015, the entirety of which is incorporated by referenceherein. PV power predictor 1212 may provide the predicted PV poweroutput {circumflex over (P)}_(PV)(k+1) to ramp rate controller 1214.

Controlling Ramp Rate

Still referring to FIG. 12, controller 814 is shown to include a ramprate controller 1214. Ramp rate controller 1214 may be configured todetermine an amount of power to charge or discharge from battery 806 forramp rate control (i.e., u_(RR)). Advantageously, ramp rate controller1214 may determine a value for the ramp rate power u_(RR) thatsimultaneously maintains the ramp rate of the PV power (i.e.,u_(RR)+P_(PV)) within compliance limits while allowing controller 814 toregulate the frequency of energy grid 812 and while maintaining thestate-of-charge of battery 806 within a predetermined desirable range.

In some embodiments, the ramp rate of the PV power is within compliancelimits as long as the actual ramp rate evaluated over a one minuteinterval does not exceed ten percent of the rated capacity of PV field802. The actual ramp rate may be evaluated over shorter intervals (e.g.,two seconds) and scaled up to a one minute interval. Therefore, a ramprate may be within compliance limits if the ramp rate satisfies one ormore of the following inequalities:

${{rr}} < {\frac{0.1\; P_{cap}}{30}\left( {1 + {tolerance}} \right)}$RR < 0.1 P_(cap)(1 + tolerance)where rr is the ramp rate calculated over a two second interval, RR isthe ramp rate calculated over a one minute interval, P_(cap) is therated capacity of PV field 802, and tolerance is an amount by which theactual ramp rate can exceed the compliance limit without resulting in anon-compliance violation (e.g., tolerance=10%). In this formulation, theramp rates rr and RR represent a difference in the PV power (e.g.,measured in kW) at the beginning and end of the ramp rate evaluationinterval.

Simultaneous implementation of ramp rate control and frequencyregulation can be challenging (e.g., can result in non-compliance),especially if the ramp rate is calculated as the difference in the powerP_(POI) at POI 810. In some embodiments, the ramp rate over a two secondinterval is defined as follows:rr=[P _(POI)(k)−P _(POI)(k−1)]−[u _(FR)(k)−u _(FR)(k−1)]where P_(POI)(k−1) and P_(POI)(k) are the total powers at POI 810measured at the beginning and end, respectively, of a two secondinterval, and u_(FR) (k−1) and u_(FR) (k) are the powers used forfrequency regulation measured at the beginning and end, respectively, ofthe two second interval.

The total power at POI 810 (i.e., P_(POI)) is the sum of the poweroutput of PV field power inverter 804 (i.e., u_(PV)) and the poweroutput of battery power inverter 808 (i.e., u_(bat)=u_(FR)+u_(RR)).Assuming that PV field power inverter 804 is not limiting the powerP_(PV) generated by PV field 802, the output of PV field power inverter804 u_(PV) may be equal to the PV power output P_(PV) (i.e.,P_(PV)=u_(PV)) and the total power P_(POI) at POI 810 can be calculatedusing the following equation:P _(POI) =P _(PV) +u _(FR) +u _(RR)Therefore, the ramp rate rr can be rewritten as:rr=P _(PV)(k)−P _(PV)(k−1)+u _(RR)(k)−u _(RR)(k−1)and the inequality which must be satisfied to comply with the ramp ratelimit can be rewritten as:

${{{P_{PV}(k)} - {P_{PV}\left( {k - 1} \right)} + {u_{RR}(k)} - {u_{RR}\left( {k - 1} \right)}}} < {\frac{0.1\; P_{cap}}{30}\left( {1 + {tolerance}} \right)}$where P_(PV)(k−1) and P_(PV)(k) are the power outputs of PV field 802measured at the beginning and end, respectively, of the two secondinterval, and u_(RR)(k−1) and u_(RR) (k) are the powers used for ramprate control measured at the beginning and end, respectively, of the twosecond interval.

In some embodiments, ramp rate controller 1214 determines the ramp ratecompliance of a facility based on the number of scans (i.e., monitoredintervals) in violation that occur within a predetermined time period(e.g., one week) and the total number of scans that occur during thepredetermined time period. For example, the ramp rate compliance RRC maybe defined as a percentage and calculated as follows:

${RRC} = {100\left( {1 - \frac{n_{vscan}}{n_{tscan}}} \right)}$where n_(vscan) is the number of scans over the predetermined timeperiod where rr is in violation and n_(tscan) is the total number ofscans during which the facility is performing ramp rate control duringthe predetermined time period.

In some embodiments, the intervals that are monitored or scanned todetermine ramp rate compliance are selected arbitrarily or randomly(e.g., by a power utility). Therefore, it may be impossible to predictwhich intervals will be monitored. Additionally, the start times and endtimes of the intervals may be unknown. In order to guarantee ramp ratecompliance and minimize the number of scans where the ramp rate is inviolation, ramp rate controller 1214 may determine the amount of poweru_(RR) used for ramp rate control ahead of time. In other words, ramprate controller 1214 may determine, at each instant, the amount of poweru_(RR) to be used for ramp rate control at the next instant. Since thestart and end times of the intervals may be unknown, ramp ratecontroller 1214 may perform ramp rate control at smaller time intervals(e.g., on the order of milliseconds).

Ramp rate controller 1214 may use the predicted PV power {circumflexover (P)}_(PV)(k+1) at instant k+1 and the current PV power P_(PV)(k) atinstant k to determine the ramp rate control power û_(RR) _(T) (k) atinstant k. Advantageously, this allows ramp rate controller 1214 todetermine whether the PV power P_(PV) is in an up-ramp, a down-ramp, orno-ramp at instant k. Assuming a T seconds time resolution, ramp ratecontroller 1214 may determine the value of the power for ramp ratecontrol û_(RR) _(T) (k) at instant k based on the predicted value of thePV power {circumflex over (P)}_(PV)(k+1), the current value of the PVpower P_(PV)(k), and the previous power used for ramp rate controlû_(RR) _(T) (k−1). Scaling to T seconds and assuming a tolerance ofzero, ramp rate compliance is guaranteed if û_(RR) _(T) (k) satisfiesthe following inequality:lb_(RR) _(T) ≤û _(RR) _(T) ≤ub_(RR) _(T)where T is the sampling time in seconds, lb_(RR) _(T) is the lower boundon û_(RR) _(T) (k), and ub_(RR) _(T) is the upper bound on û_(RR) _(T)(k).

In some embodiments, the lower bound lb_(RR) _(T) and the upper boundub_(RR) _(T) are defined as follows:

${lb}_{{RR}_{T}} = {{- \left( {{{\hat{P}}_{PV}\left( {k + 1} \right)} - {P_{PV}(k)}} \right)} + {{\hat{u}}_{{RR}_{T}}\left( {k - 1} \right)} - \frac{0.1\; P_{cap}}{60/T} + {\lambda\;\sigma}}$${ub}_{{RR}_{T}} = {{- \left( {{{\hat{P}}_{PV}\left( {k + 1} \right)} - {P_{PV}(k)}} \right)} + {{\hat{u}}_{{RR}_{T}}\left( {k - 1} \right)} + \frac{0.1\; P_{cap}}{60/T} - {\lambda\;\sigma}}$where σ is the uncertainty on the PV power prediction and λ is a scalingfactor of the uncertainty in the PV power prediction. Advantageously,the lower bound lb_(RR) _(T) and the upper bound ub_(RR) _(T) provide arange of ramp rate power û_(RR) _(T) (k) that guarantees compliance ofthe rate of change in the PV power.

In some embodiments, ramp rate controller 1214 determines the ramp ratepower û_(RR) _(T) (k) based on whether the PV power P_(PV) is in anup-ramp, a down-ramp, or no-ramp (e.g., the PV power is not changing orchanging at a compliant rate) at instant k. Ramp rate controller 1214may also consider the state-of-charge (SOC) of battery 806 whendetermining û_(RR) _(T) (k). Exemplary processes which may be performedby ramp rate controller 1214 to generate values for the ramp rate powerû_(RR) _(T) (k) are described in detail in U.S. Patent Application No.62/239,245. Ramp rate controller 1214 may provide the ramp rate powersetpoint û_(RR) _(T) (k) to battery power setpoint generator 1218 foruse in determining the battery power setpoint u_(bat).

Controlling Frequency Regulation

Referring again to FIG. 12, controller 814 is shown to include afrequency regulation controller 1216. Frequency regulation controller1216 may be configured to determine an amount of power to charge ordischarge from battery 806 for frequency regulation (i.e., u_(FR)).Frequency regulation controller 1216 is shown receiving a grid frequencysignal from energy grid 812. The grid frequency signal may specify thecurrent grid frequency f_(grid) of energy grid 812. In some embodiments,the grid frequency signal also includes a scheduled or desired gridfrequency f_(s) to be achieved by performing frequency regulation.Frequency regulation controller 1216 may determine the frequencyregulation setpoint u_(FR) based on the difference between the currentgrid frequency f_(grid) and the scheduled frequency f_(s).

In some embodiments, the range within which the grid frequency f_(grid)is allowed to fluctuate is determined by an electric utility. Anyfrequencies falling outside the permissible range may be corrected byperforming frequency regulation. Facilities participating in frequencyregulation may be required to supply or consume a contracted power forpurposes of regulating grid frequency f_(grid) (e.g., up to 10% of therated capacity of PV field 802 per frequency regulation event).

In some embodiments, frequency regulation controller 1216 performsfrequency regulation using a dead-band control technique with a gainthat is dependent upon the difference f_(e) between the scheduled gridfrequency f_(s) and the actual grid frequency f_(grid) (i.e.,f_(e)=f_(s)−f_(grid)) and an amount of power required for regulating agiven deviation amount of frequency error f_(e). Such a controltechnique is expressed mathematically by the following equation:u _(FR)(k)=min(max(lb_(FR),α),ub_(FR))where lb_(FR) and ub_(FR) are the contracted amounts of power up towhich power is to be consumed or supplied by a facility. lb_(FR) andub_(FR) may be based on the rated capacity P_(cap) of PV field 802 asshown in the following equations:lb_(FR)=−0.1×P _(cap)ub_(FR)=0.1×P _(cap)

The variable α represents the required amount of power to be supplied orconsumed from energy grid 812 to offset the frequency error f_(e). Insome embodiments, frequency regulation controller 1216 calculates ausing the following equation:α=K _(FR)×sign(f _(e))×max(|f _(e) |−d _(band),0)where d_(band) is the threshold beyond which a deviation in gridfrequency must be regulated and K_(FR) is the control gain. In someembodiments, frequency regulation controller 1216 calculates the controlgain K_(FR) as follows:

$K_{FR} = \frac{P_{cap}}{0.01 \times {droop} \times f_{s}}$where droop is a parameter specifying a percentage that defines how muchpower must be supplied or consumed to offset a 1 Hz deviation in thegrid frequency. Frequency regulation controller 1216 may calculate thefrequency regulation setpoint u_(FR) using these equations and mayprovide the frequency regulation setpoint to battery power setpointgenerator 1218.Generating Battery Power Setpoints

Still referring to FIG. 12, controller 814 is shown to include a batterypower setpoint generator 1218. Battery power setpoint generator 1218 maybe configured to generate the battery power setpoint u_(bat) for batterypower inverter 808. The battery power setpoint u_(bat) is used bybattery power inverter 808 to control an amount of power drawn frombattery 806 or stored in battery 806. For example, battery powerinverter 808 may draw power from battery 806 in response to receiving apositive battery power setpoint u_(bat) from battery power setpointgenerator 1218 and may store power in battery 806 in response toreceiving a negative battery power setpoint u_(bat) from battery powersetpoint generator 1218.

Battery power setpoint generator 1218 is shown receiving the ramp ratepower setpoint u_(RR) from ramp rate controller 1214 and the frequencyregulation power setpoint u_(FR) from frequency regulation controller1216. In some embodiments, battery power setpoint generator 1218calculates a value for the battery power setpoint u_(bat) by adding theramp rate power setpoint u_(RR) and the frequency response powersetpoint u_(FR). For example, battery power setpoint generator 1218 maycalculate the battery power setpoint u_(bat) using the followingequation:u _(bat) =u _(RR) +u _(FR)

In some embodiments, battery power setpoint generator 1218 adjusts thebattery power setpoint u_(bat) based on a battery power limit forbattery 806. For example, battery power setpoint generator 1218 maycompare the battery power setpoint u_(bat) with the battery power limitbattPowerLimit. If the battery power setpoint is greater than thebattery power limit (i.e., u_(bat)>battPowerLimit), battery powersetpoint generator 1218 may replace the battery power setpoint u_(bat)with the battery power limit. Similarly, if the battery power setpointis less than the negative of the battery power limit (i.e.,u_(bat)<−battPowerLimit), battery power setpoint generator 1218 mayreplace the battery power setpoint u_(bat) with the negative of thebattery power limit.

In some embodiments, battery power setpoint generator 1218 causesfrequency regulation controller 1216 to update the frequency regulationsetpoint u_(FR) in response to replacing the battery power setpointu_(bat) with the battery power limit battPowerLimit or the negative ofthe battery power limit −battPowerLimit. For example, if the batterypower setpoint u_(bat) is replaced with the positive battery power limitbattPowerLimit, frequency regulation controller 1216 may update thefrequency regulation setpoint u_(FR) using the following equation:u _(FR)(k)=battPowerLimit−û _(RR) _(T) (k)

Similarly, if the battery power setpoint u_(bat) is replaced with thenegative battery power limit −battPowerLimit, frequency regulationcontroller 1216 may update the frequency regulation setpoint u_(FR)using the following equation:u _(FR)(k)=−battPowerLimit−û _(RR) _(T) (k)These updates ensure that the amount of power used for ramp rate controlû_(RR) _(T) (k) and the amount of power used for frequency regulationu_(FR) (k) can be added together to calculate the battery power setpointu_(bat). Battery power setpoint generator 1218 may provide the batterypower setpoint u_(bat) to battery power inverter 808 and to PV powersetpoint generator 1220.Generating PV Power Setpoints

Still referring to FIG. 12, controller 814 is shown to include a PVpower setpoint generator 1220. PV power setpoint generator 1220 may beconfigured to generate the PV power setpoint u_(PV) for PV field powerinverter 804. The PV power setpoint u_(PV) is used by PV field powerinverter 804 to control an amount of power from PV field 802 to provideto POI 810.

In some embodiments, PV power setpoint generator 1220 sets a default PVpower setpoint u_(PV)(k) for instant k based on the previous value ofthe PV power P_(PV)(k−1) at instant k−1. For example, PV power setpointgenerator 1220 may increment the previous PV power P_(PV)(k−1) with thecompliance limit as shown in the following equation:

${u_{PV}(k)} = {{P_{PV}\left( {k - 1} \right)} + \frac{0.1\; P_{cap}}{60/T} - {\lambda\;\sigma}}$This guarantees compliance with the ramp rate compliance limit andgradual ramping of the PV power output to energy grid 812. The defaultPV power setpoint may be useful to guarantee ramp rate compliance whenthe system is turned on, for example, in the middle of a sunny day orwhen an up-ramp in the PV power output P_(PV) is to be handled bylimiting the PV power at PV power inverter 804 instead of chargingbattery 806.

In some embodiments, PV power setpoint generator 1220 updates the PVpower setpoint u_(PV)(k) based on the value of the battery powersetpoint u_(bat)(k) so that the total power provided to POI 810 does notexceed a POI power limit. For example, PV power setpoint generator 1220may use the PV power setpoint u_(PV)(k) and the battery power setpointu_(bat)(k) to calculate the total power P_(POI)(k) at point ofintersection 810 using the following equation:P _(POI)(k)=u _(bat)(k)+u _(PV)(k)

PV power setpoint generator 1220 may compare the calculated powerP_(POI)(k) with a power limit for POI 810 (i.e., POIPowerLimit). If thecalculated power P_(POI)(k) exceeds the POI power limit (i.e.,P_(POI)(k)>POIPowerLimit), PV power setpoint generator 1220 may replacethe calculated power P_(POI)(k) with the POI power limit. PV powersetpoint generator 1220 may update the PV power setpoint u_(PV)(k) usingthe following equation:u _(PV)(k)=POIPowerLimit−u _(bat)(k)This ensures that the total power provided to POI 810 does not exceedthe POI power limit by causing PV field power inverter 804 to limit thePV power. PV power setpoint generator 1220 may provide the PV powersetpoint u_(PV) to PV field power inverter 804.Frequency Response Control System

Referring now to FIG. 13, a block diagram of a frequency responsecontrol system 1300 is shown, according to exemplary embodiment. Controlsystem 1300 is shown to include frequency response controller 112, whichmay be the same or similar as previously described. For example,frequency response controller 112 may be configured to perform anoptimization process to generate values for the bid price, thecapability bid, and the midpoint b. In some embodiments, frequencyresponse controller 112 generates values for the bids and the midpoint bperiodically using a predictive optimization scheme (e.g., once everyhalf hour, once per frequency response period, etc.). Frequency responsecontroller 112 may also calculate and update power setpoints for powerinverter 106 periodically during each frequency response period (e.g.,once every two seconds). As shown in FIG. 13, frequency responsecontroller 112 is in communication with one or more external systems viacommunication interface 1302. Additionally, frequency responsecontroller 112 is also shown as being in communication with a batterysystem 1304.

In some embodiments, the interval at which frequency response controller112 generates power setpoints for power inverter 106 is significantlyshorter than the interval at which frequency response controller 112generates the bids and the midpoint b. For example, frequency responsecontroller 112 may generate values for the bids and the midpoint b everyhalf hour, whereas frequency response controller 112 may generate apower setpoint for power inverter 106 every two seconds. The differencein these time scales allows frequency response controller 112 to use acascaded optimization process to generate optimal bids, midpoints b, andpower setpoints.

In the cascaded optimization process, high level controller 312determines optimal values for the bid price, the capability bid, and themidpoint b by performing a high level optimization. The high levelcontroller 312 may be a centralized server within the frequency responsecontroller 112. The high level controller 312 may be configured toexecute optimization control algorithms, such as those described herein.In one embodiment, the high level controller 312 may be configured torun an optimization engine, such as a MATLAB optimization engine.

Further, the cascaded optimization process allows for multiplecontrollers to process different portions of the optimization process.As will be described below, the high level controller 312 may be used toperform optimization functions based on received data, while a low levelcontroller 314 may receive optimization data from the high levelcontroller 312 and control the battery system 1304 accordingly. Byallowing independent platforms to perform separation portions of theoptimization, the individual platforms may be scaled and tunedindependently. For example, the controller 112 may be able to be scaledup to accommodate a larger battery system 1304 by adding additional lowlevel controllers to control the battery system 1304. Further, the highlevel controller 312 may be modified to provide additional computingpower for optimizing battery system 1304 in more complex systems.Further, modifications to either the high level controller 312 or thelow level controller 314 will not affect the other, thereby increasingoverall system stability and availability.

In system 1300, high level controller 312 may be configured to performsome or all of the functions previously described with reference toFIGS. 3-4. For example, high level controller 312 may select midpoint bto maintain a constant state-of-charge in battery 108 (i.e., the samestate-of-charge at the beginning and end of each frequency responseperiod) or to vary the state-of-charge in order to optimize the overallvalue of operating system 1300 (e.g., frequency response revenue minusenergy costs and battery degradation costs), as described below. Highlevel controller 312 may also determine filter parameters for a signalfilter (e.g., a low pass filter) used by a low level controller 314.

The low level controller 314 may be a standalone controller. In oneembodiment, the low level controller 314 is a Network Automation Engine(NAE) controller from Johnson Controls. However, other controllershaving the required capabilities are also contemplated. The requiredcapabilities for the low level controller 314 may include havingsufficient memory and computing power to run the applications, describedbelow, at the required frequencies. For example, certain optimizationcontrol loops (described below) may require control loops running at 200ms intervals. However, intervals of more than 200 ms and less than 200ms may also be required. These control loops may require reading andwriting data to and from the battery inverter. The low level controller314 may also be required to support Ethernet connectivity (or othernetwork connectivity) to connect to a network for receiving bothoperational data, as well as configuration data. The low levelcontroller 314 may be configured to perform some or all of the functionspreviously described with reference to FIGS. 3-5.

The low level controller 314 may be capable of quickly controlling oneor more devices around one or more setpoints. For example, low levelcontroller 314 uses the midpoint b and the filter parameters from highlevel controller 312 to perform a low level optimization in order togenerate the power setpoints for power inverter 106. Advantageously, lowlevel controller 314 may determine how closely to track the desiredpower P*_(POI) at the point of interconnection 110. For example, the lowlevel optimization performed by low level controller 314 may considernot only frequency response revenue but also the costs of the powersetpoints in terms of energy costs and battery degradation. In someinstances, low level controller 314 may determine that it is deleteriousto battery 108 to follow the regulation exactly and may sacrifice aportion of the frequency response revenue in order to preserve the lifeof battery 108.

Low level controller 314 may also be configured to interface with one ormore other devises or systems. For example, the low level controller 314may communicate with the power inverter 106 and/or the batterymanagement unit 1310 via a low level controller communication interface1312. Communications interface 1312 may include wired or wirelessinterfaces (e.g., jacks, antennas, transmitters, receivers,transceivers, wire terminals, etc.) for conducting data communicationswith various systems, devices, or networks. For example, communicationsinterface 1312 may include an Ethernet card and port for sending andreceiving data via an Ethernet-based communications network and/or aWiFi transceiver for communicating via a wireless communicationsnetwork. Communications interface 1312 may be configured to communicatevia local area networks or wide area networks (e.g., the Internet, abuilding WAN, etc.) and may use a variety of communications protocols(e.g., BACnet, MODBUS, CAN, IP, LON, etc.).

As described above, the low level controller 314 may communicatesetpoints to the power inverter 106. Furthermore, the low levelcontroller 314 may receive data from the battery management unit 1310via the communication interface 1312. The battery management unit 1310may provide data relating to a state of charge (SOC) of the batteries108. The battery management unit 1310 may further provide data relatingto other parameters of the batteries 108, such as temperature, real timeor historical voltage level values, real time or historical currentvalues, etc. The low level controller 314 may be configured to performtime critical functions of the frequency response controller 112. Forexample, the low level controller 314 may be able to perform fast loop(PID, PD, PI, etc.) controls in real time.

The low level controller 314 may further control a number of othersystems or devices associated with the battery system 1304. For example,the low level controller may control safety systems 1316 and/orenvironmental systems 1318. In one embodiment, the low level controller314 may communicate with and control the safety systems 1316 and/or theenvironmental systems 1318 through an input/output module (IOM) 1319. Inone example, the IOM may be an IOM controller from Johnson Controls. TheIOM may be configured to receive data from the low level controller andthen output discrete control signals to the safety systems 1316 and/orenvironmental systems 1318. Further, the IOM 1319 may receive discreteoutputs from the safety systems 1316 and/or environmental systems 320,and report those values to the low level controller 314. For example,the IOM 1319 may provide binary outputs to the environmental system1318, such as a temperature setpoint; and in return may receive one ormore analog inputs corresponding to temperatures or other parametersassociated with the environmental systems 1318. Similarly, the safetysystems 1316 may provide binary inputs to the IOM 1319 indicating thestatus of one or more safety systems or devices within the batterysystem 1304. The IOM 1319 may be able to process multiple data pointsfrom devices within the battery system 1304. Further, the IOM may beconfigured to receive and output a variety of analog signals (4-20 mA,0-5V, etc.) as well as binary signals.

The environmental systems 1318 may include HVAC devices such as roof-topunits (RTUs), air handling units (AHUs), etc. The environmental systems1318 may be coupled to the battery system 1304 to provide environmentalregulation of the battery system 1304. For example, the environmentalsystems 1318 may provide cooling for the battery system 1304. In oneexample, the battery system 1304 may be contained within anenvironmentally sealed container. The environmental systems 1318 maythen be used to not only provide airflow through the battery system1304, but also to condition the air to provide additional cooling to thebatteries 108 and/or the power inverter 106. The environmental systems1318 may also provide environmental services such as air filtration,liquid cooling, heating, etc. The safety systems 1316 may providevarious safety controls and interlocks associated with the batterysystem 1304. For example, the safety systems 1316 may monitor one ormore contacts associated with access points on the battery system. Wherea contact indicates that an access point is being accessed, the safetysystems 1316 may communicate the associated data to the low levelcontroller 314 via the IOM 1319. The low level controller may thengenerate and alarm and/or shut down the battery system 1304 to preventany injury to a person accessing the battery system 1304 duringoperation. Further examples of safety systems can include air qualitymonitors, smoke detectors, fire suppression systems, etc.

Still referring to FIG. 13, the frequency response controller 112 isshown to include the high level controller communications interface1302. Communications interface 1302 may include wired or wirelessinterfaces (e.g., jacks, antennas, transmitters, receivers,transceivers, wire terminals, etc.) for conducting data communicationswith various systems, devices, or networks. For example, communicationsinterface 1302 may include an Ethernet card and port for sending andreceiving data via an Ethernet-based communications network and/or aWiFi transceiver for communicating via a wireless communicationsnetwork. Communications interface 1302 may be configured to communicatevia local area networks or wide area networks (e.g., the Internet, abuilding WAN, etc.) and may use a variety of communications protocols(e.g., BACnet, IP, LON, etc.).

Communications interface 1302 may be a network interface configured tofacilitate electronic data communications between frequency responsecontroller 112 and various external systems or devices (e.g., campus102, energy grid 104, incentive provider 114, utilities 320, weatherservice 322, etc.). For example, frequency response controller 112 mayreceive inputs from incentive provider 114 indicating an incentive eventhistory (e.g., past clearing prices, mileage ratios, participationrequirements, etc.) and a regulation signal. Further, the incentiveprovider 114 may communicate utility rates provided by utilities 320.Frequency response controller 112 may receive a campus power signal fromcampus 102, and weather forecasts from weather service 322 viacommunications interface 1302. Frequency response controller 112 mayprovide a price bid and a capability bid to incentive provider 114 andmay provide power setpoints to power inverter 106 via communicationsinterface 1302.

Data Fusion

Turning now to FIG. 14, a block diagram illustrating data flow into thedata fusion module 428 is shown, according to some embodiments. As shownin FIG. 14, the data fusion module 428 may receive data from multipledevices and/or systems. In one embodiment, the data fusion module 428may receive all data received by the high level controller 312. Forexample, the data fusion module 428 may receive campus data from thecampus 102. Campus data may include campus power requirements, campuspower requests, occupancy planning, historical use data, lightingschedules, HVAC schedules, etc. In a further embodiment, the data fusionmodule 428 may receive weather data from the weather service 322. Theweather service 322 may include current weather data (temperature,humidity, barometric pressure, etc.), weather forecasts, historicalweather data, etc. In a still further embodiment, the data fusion module428 may receive utility data from the utilities 320. In some examples,the data fusion module 428 may receive some or all of the utility datavia the incentive provider 114. Examples of utility data may includeutility rates, future pricing schedules, anticipated loading, historicaldata, etc. Further, the incentive provider 114 may further add data suchas capability bid requests, price bid requests, incentive data, etc.

The data fusion module 428 may further receive data from the low levelcontroller 314. In some embodiments, the low level controller mayreceive data from multiple sources, which may be referred tocollectively as battery system data. For example, the low levelcontroller 314 may receive inverter data from power inverter 106.Example inverter data may include inverter status, feedback points,inverter voltage and current, power consumption, etc. The low levelcontroller 314 may further receive battery data from the batterymanagement unit 1310. Example battery data may include battery SOC,depth of discharge data, battery temperature, battery cell temperatures,battery voltage, historical battery use data, battery health data, etc.In other embodiment, the low level controller 314 may receiveenvironmental data from the environmental systems 1318. Examples ofenvironmental data may include battery system temperature, batterysystem humidity, current HVAC settings, setpoint temperatures,historical HVAC data, etc. Further, the low level controller 314 mayreceive safety system data from the safety systems 1316. Safety systemdata may include access contact information (e.g. open or closedindications), access data (e.g. who has accessed the battery system 1304over time), alarm data, etc. In some embodiments, some or all of thedata provided to the low level controller 314 is via an input/outputmodule, such as IOM 1319. For example, the safety system data and theenvironmental system data may be provided to the low level controller314 via an input/output module, as described in detail in regards toFIG. 13.

The low level controller 314 may then communicate the battery systemdata to the data fusion module 428 within the high level controller 312.Additionally, the low level controller 314 may provide additional datato the data fusion module 428, such as setpoint data, controlparameters, etc.

The data fusion module 428 may further receive data from otherstationary power systems, such as a photovoltaic system 1402. Forexample, the photovoltaic system 1402 may include one or morephotovoltaic arrays and one or more photovoltaic array power inverters.The photovoltaic system 1402 may provide data to the data fusion module428 such as photovoltaic array efficiency, photovoltaic array voltage,photovoltaic array inverter output voltage, photovoltaic array inverteroutput current, photovoltaic array inverter temperature, etc. In someembodiments, the photovoltaic system 1402 may provide data directly tothe data fusion module 428 within the high level controller 312. Inother embodiments, the photovoltaic system 1402 may transmit the data tothe low level controller 314, which may then provide the data to thedata fusion module 428 within the high level controller 312.

The data fusion module 428 may receive some or all of the data describedabove, and aggregate the data for use by the high level controller 312.In one embodiment, the data fusion module 428 is configured to receiveand aggregate all data received by the high level controller 312, and tosubsequently parse and distribute the data to one or more modules of thehigh level controller 312, as described above. Further, the data fusionmodule 428 may be configured to combine disparate heterogeneous datafrom the multiple sources described above, into a homogeneous datacollection for use by the high level controller 312. As described above,data from multiple inputs is required to optimize the battery system1304, and the data fusion module 428 can gather and process the datasuch that it can be provided to the modules of the high level controller312 efficiently and accurately. For example, extending battery lifespanis critical for ensuring proper utilization of the battery system 1304.By combining battery data such as temperature and voltage, along withexternal data such as weather forecasts, remaining battery life may bemore accurately determined by the battery degradation estimator 418,described above. Similarly, multiple data points from both externalsources and the battery system 1304 may allow for more accurate midpointestimations, revenue loss estimations, battery power loss estimation, orother optimization determination, as described above.

Turning now to FIG. 15, a block diagram showing a database schema 1500of the system 1300 is shown, according to some embodiments. The schema1500 is shown to include an algorithm run data table 1502, a data pointdata table 1504, an algorithm_run time series data table 1508 and apoint time series data table 1510. The data tables 1502, 1504, 1508,1510 may be stored on the memory of the high level controller 312. Inother embodiments, the data tables 1502, 1504, 1508, 1510 may be storedon an external storage device and accessed by the high level controlleras required.

As described above, the high level controller performs calculation togenerate optimization data for the battery optimization system 1300.These calculation operations (e.g. executed algorithms) may be referredto as “runs.” As described above, one such run is the generation of amidpoint b which can subsequently be provided to the low levelcontroller 314 to control the battery system 1304. However, other typesof runs are contemplated. Thus, for the above described run, themidpoint b is the output of the run. The detailed operation of a run,and specifically a run to generate midpoint b is described in detailabove.

The algorithm run data table 1502 may include a number of algorithm runattributes 1512. Algorithm run attributes 1512 are those attributesassociated with the high level controller 312 executing an algorithm, or“run”, to produce an output. The runs can be performed at selectedintervals of time. For example, the run may be performed once everyhour. However, in other examples, the run may be performed more thanonce every hour, or less than once every hour. The run is then performedand by the high level controller 312 and a data point is output, forexample a midpoint b, as described above. The midpoint b may be providedto the low level controller 314 to control the battery system 1304,described above in the description of the high level controller 1304calculating the midpoint b.

In one embodiment, the algorithm run attributes contain all theinformation necessary to perform the algorithm or run. In a furtherembodiment, the algorithm run attributes 1512 are associated with thehigh level controller executing an algorithm to generate a midpoint,such as midpoint b described in detail above. Example algorithm runattributes may include an algorithm run key, an algorithm run ID (e.g.“midpoint,” “endpoint,” “temperature_setpoint,” etc.), Associated Run ID(e.g. name of the run), run start time, run stop time, target run time(e.g. when is the next run desired to start), run status, run reason,fail reason, plant object ID (e.g. name of system), customer ID, runcreator ID, run creation date, run update ID, and run update date.However, this list is for example only, as it is contemplated that thealgorithm run attributes may contain multiple other attributesassociated with a given run.

As stated above, the algorithm run data table 1502 contains attributesassociated with a run to be performed by the high level controller 312.In some embodiments, the output of a run, is one or more “points,” suchas a midpoint. The data point data table 1504 contains data pointattributes 1514 associated with various points that may be generated bya run. These data point attributes 1514 are used to describe thecharacteristics of the data points. For example, the data pointattributes may contain information associated with a midpoint datapoint. However, other data point types are contemplated. Exampleattributes may include point name, default precision (e.g. number ofsignificant digits), default unit (e.g. cm, degrees Celsius, voltage,etc.), unit type, category, fully qualified reference (yes or no),attribute reference ID, etc. However, other attributes are furtherconsidered.

The algorithm_run time series data table 1508 may contain time seriesdata 1516 associated with a run. In one embodiment, the algorithm_runtime series data 1516 includes time series data associated with aparticular algorithm run ID. For example, a run associated withdetermining the midpoint b described above, may have an algorithm run IDof Midpoint_Run. The algorithm_run time series data table 1508 maytherefore include algorithm_run time series data 1516 for all runsperformed under the algorithm ID Midpoint_Run. Additionally, thealgorithm_run time series data table 1508 may also contain run timeseries data associated with other algorithm IDs as well. The run timeseries data 1516 may include past data associated with a run, as well asexpected future information. Example run time series data 1516 mayinclude final values of previous runs, the unit of measure in theprevious runs, previous final value reliability values, etc. As anexample, a “midpoint” run may be run every hour, as described above. Thealgorithm_run time series data 1516 may include data related to thepreviously performed runs, such as energy prices over time, system data,etc. Additionally, the algorithm_run time series data 1516 may includepoint time series data associated with a given point, as describedbelow.

The point time series data table 1510 may include the point time seriesdata 1518. The point time series data 1518 may include time series dataassociated with a given data “point.” For example, the above describedmidpoint b may have a point ID of “Midpoint.” The point time series datatable 1510 may contain point time series data 1518 associated with the“midpoint” ID, generated over time. For example, previous midpointvalues may be stored in the point time series data table 1518 for eachperformed run. The point time series data table 1510 may identify theprevious midpoint values by time (e.g. when the midpoint was used by thelow level controller 314), and may include information such as themidpoint value, reliability information associated with the midpoint,etc. In one embodiment, the point time series data table 1518 may beupdated with new values each time a new “midpoint” is generated via arun. Further, the point time series data 1516 for a given point mayinclude information independent of a given run. For example, the highlevel controller 312 may monitor other data associated with themidpoint, such as regulation information from the low level controller,optimization data, etc., which may further be stored in the point timeseries data table 1510 as point time series data 1518.

The above described data tables may be configured to have an associationor relational connection between them. For example, as shown in FIG. 15,the algorithm_run data table 1502 may have a one-to-many association orrelational relationship with the algorithm_run time series associationtable 1508, as there may be many algorithm_run time series data points1516 for each individual algorithm run ID. Further, the data point datatable 1504 may have a one-to many relationship with the point timeseries data table 1510, as there may be many point time series datapoints 1518 associated with an individual point. Further, the point timeseries data table 1510 may have a one to many relationship with thealgorithm_run time series data table 1508, as there may be multipledifferent point time series data 1518 associated with a run.Accordingly, the algorithm_run data table 1502 has a many-to-manyrelationship with the data point data table 1504, as there may be manypoints, and/or point time series data 1518, associated with may runtypes; and, there may be multiple run types associated with many points

By using the above mentioned association data tables 1502, 1504, 1508,1510, optimization of storage space required for storing time seriesdata may be achieved. With the addition of additional data used in abattery optimization system, such as battery optimization system 1300described above, vast amounts of time series data related to dataprovided by external sources (weather data, utility data, campus data,building automation systems (BAS) or building management systems (BMS)),and internal sources (battery systems, photovoltaic systems, etc.) isgenerated. By utilizing association data tables, such as those describedabove, the data may be optimally stored and accessed.

Configuration of Exemplary Embodiments

The construction and arrangement of the systems and methods as shown inthe various exemplary embodiments are illustrative only. Although only afew embodiments have been described in detail in this disclosure, manymodifications are possible (e.g., variations in sizes, dimensions,structures, shapes and proportions of the various elements, values ofparameters, mounting arrangements, use of materials, colors,orientations, etc.). For example, the position of elements can bereversed or otherwise varied and the nature or number of discreteelements or positions can be altered or varied. Accordingly, all suchmodifications are intended to be included within the scope of thepresent disclosure. The order or sequence of any process or method stepscan be varied or re-sequenced according to alternative embodiments.Other substitutions, modifications, changes, and omissions can be madein the design, operating conditions and arrangement of the exemplaryembodiments without departing from the scope of the present disclosure.

The present disclosure contemplates methods, systems and programproducts on any machine-readable media for accomplishing variousoperations. The embodiments of the present disclosure can be implementedusing existing computer processors, or by a special purpose computerprocessor for an appropriate system, incorporated for this or anotherpurpose, or by a hardwired system. Embodiments within the scope of thepresent disclosure include program products comprising machine-readablemedia for carrying or having machine-executable instructions or datastructures stored thereon. Such machine-readable media can be anyavailable media that can be accessed by a general purpose or specialpurpose computer or other machine with a processor. By way of example,such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROMor other optical disk storage, magnetic disk storage or other magneticstorage devices, or any other medium which can be used to carry or storedesired program code in the form of machine-executable instructions ordata structures and which can be accessed by a general purpose orspecial purpose computer or other machine with a processor. Combinationsof the above are also included within the scope of machine-readablemedia. Machine-executable instructions include, for example,instructions and data which cause a general purpose computer, specialpurpose computer, or special purpose processing machines to perform acertain function or group of functions.

Although the figures show a specific order of method steps, the order ofthe steps may differ from what is depicted. Also two or more steps canbe performed concurrently or with partial concurrence. Such variationwill depend on the software and hardware systems chosen and on designerchoice. All such variations are within the scope of the disclosure.Likewise, software implementations could be accomplished with standardprogramming techniques with rule based logic and other logic toaccomplish the various connection steps, processing steps, comparisonsteps and decision steps.

What is claimed is:
 1. An electrical energy storage system comprising: abattery configured to store electric power and discharge electric powerto an energy grid; a power inverter configured to control an amount ofthe electric power stored to or discharged from the battery; and acontroller configured to determine battery power setpoints for the powerinverter based on a battery life model, the battery life model includingat least one variable that depends on the battery power setpoints,determine an amount of battery degradation for the battery based on theat least one variable, estimate an amount of frequency response revenueto be lost due to the amount of battery degradation for the battery, andmodify the battery power setpoints based on the estimated amount offrequency response revenue to be lost due to the amount of batterydegradation for the battery.
 2. The system of claim 1, wherein theamount of battery degradation includes an amount of battery capacityloss.
 3. The system of claim 1, wherein the battery power setpointsinclude frequency regulation power setpoints and ramp rate control powersetpoints.
 4. The system of claim 3, wherein the frequency regulationpower setpoints are based on a frequency of the energy grid; and whereinthe ramp rate control power setpoints are based on a power output of aphotovoltaic field.
 5. The system of claim 1, wherein the at least onevariable is selected from the group consisting of a temperature of thebattery, a state-of-charge of the battery, a depth of discharge of thebattery, a power ratio of the battery, and an effort ratio of thebattery.
 6. The system of claim 1, wherein the controller is furtherconfigured to estimate an amount of frequency response revenue to begained based on the battery power setpoints.
 7. The system of claim 6,wherein the controller is further configured to modify the battery powersetpoints as a function of the amount of frequency response revenue tobe gained based on the battery power setpoints.
 8. The system of claim1, wherein the battery life model is a parametric model including aregression coefficient for the at least one variable of the battery lifemodel.
 9. The system of claim 8, wherein the controller is furtherconfigured to perform a curve fitting process to determine a value forthe regression coefficient.
 10. A method of operating an electricalenergy storage system, the method comprising: determining battery powersetpoints for a power inverter based on a battery life model, thebattery life model including at least one variable that depends on thebattery power setpoints; determining an amount of battery degradationfor the battery based on the at least one variable; estimating an amountof frequency response revenue to be lost due to the amount of batterydegradation for the battery; and modifying the battery power setpointsbased on the estimated amount of frequency response revenue to be lostdue to the amount of battery degradation for the battery.
 11. The methodof claim 10, wherein the amount of battery degradation includes anamount of battery capacity loss.
 12. The method of claim 10, wherein thebattery power setpoints include frequency regulation power setpoints andramp rate control power setpoints.
 13. The method of claim 12, whereinthe frequency regulation power setpoints are based on a frequency of anenergy grid; and wherein the ramp rate control power setpoints are basedon a power output of a photovoltaic field.
 14. The method of claim 10,wherein the at least one variable is selected from the group consistingof a temperature of the battery, a state-of-charge of the battery, adepth of discharge of the battery, a power ratio of the battery, and aneffort ratio of the battery.
 15. The method of claim 10, furthercomprising estimating an amount of frequency response revenue to begained based on the battery power setpoints.
 16. The method of claim 15,further comprising modifying the battery power setpoints as a functionof the amount of frequency response revenue to be gained based on thebattery power setpoints.
 17. A controller for an electrical energystorage system, the controller including a computer readable medium anda processor, the controller comprising computer executable instructionsstored in the computer readable medium for controlling the electricalenergy storage system to: determine battery power setpoints for thepower inverter based on a battery life model, the battery life modelincluding at least one variable that depends on the battery powersetpoints; determine an amount of battery degradation for the batterybased on the at least one variable; estimate an amount of frequencyresponse revenue to be lost due to the amount of battery degradation forthe battery; and modify the battery power setpoints based on theestimated amount of frequency response revenue to be lost due to theamount of battery degradation for the battery.
 18. The controller ofclaim 17, the controller further comprising computer executableinstructions stored in the computer readable medium for controlling theelectrical energy storage system to: estimate an amount of frequencyresponse revenue to be gained based on the battery power setpoints. 19.The controller of claim 18, the controller further comprising computerexecutable instructions stored in the computer readable medium forcontrolling the electrical energy storage system to: modify the batterypower setpoints as a function of the amount of frequency responserevenue to be gained based on the battery power setpoints.
 20. Thecontroller of claim 17, wherein the amount of battery degradationincludes an amount of battery capacity loss.